Control of single spin in Markovian environment
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Yuan, Haidong. "Control of single spin in Markovian environment ."
Proceedings of the Joint 48th IEEE Conference on Decision and
Control and 28th Chinese Control Conference Shanghai, P.R. China,
December 16-18, 2009.
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http://dx.doi.org/10.1109/CDC.2009.5400712
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Institute of Electrical and Electronics Engineers
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Final published version
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http://hdl.handle.net/1721.1/74601
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Control of single spin in Markovian environment
Haidong Yuan
∗Abstract
In this article we study the control of single spin in Markovian environment. Given an initial state, we com-pute all the possible states to which the spin can be driven at arbitrary time, under the assumption that fast unitary operations on the single spin are available.
1. Introduction
In the last two decades, control theory has been ap-plied to an increasingly wide number of problems in physics and chemistry whose dynamics are governed by the time-dependent Schr¨odinger equation (TDSE), including control of chemical reactions [1, 2, 3, 5, 6, 7, 8, 9], state-to-state population transfer [10, 11, 12, 13], shaped wavepackets [14], NMR spin dynamics [15], Bose-Einstein condensation [16, 17, 18], quantum com-puting [19, 20, 21], oriented rotational wavepackets [22], etc. [23, 24]. More recently, there has been vigor-ous effort in studying the control of open quantum sys-tems which are governed by Lindblad equations, where the central object is the density matrix, rather than the wavefunction [25, 26, 27, 28, 29, 30, 31]. The Lind-blad equation is an extension of the TDSE that allows for the inclusion of dissipative processes. In this arti-cle, we study two level systems governed by the con-trolled Lindblad equation, we will compute all the pos-sible states to which the systems can be driven at arbi-trary time, under the assumption that fast unitary oper-ations on the system are available.
2. Setting up the control problem
2.1. The system equations of motion and the
Lindblad formula for dissipation
Let ρ denote the density matrix of an quantum sys-tem. The density matrix evolves under the Lindblad
∗Haidong Yuan is with the department of Mechanical Engineering,
Massachusetts Institute of Technology.
equation, which takes the form ˙
ρ = −i[H(t), ρ ] + L(ρ ) (1) where −i[H, ρ] is the unitary evolution of the quantum system and L(ρ) is the dissipative part of the evolution. The term L(ρ) is linear in ρ and is given by the Lindblad form [32, 34], L(ρ) =
∑
i j aα β(Fαρ Fβ†− 1 2{F † βFα, ρ}),where Fα, Fβ are the Lindblad operators. Eq. (1) has
the following three well known properties: 1) Tr(ρ) re-mains unity for all time, 2) ρ rere-mains a Hermitian ma-trix, and 3) ρ stays positive semi-positive definite, i.e. that ρ never develops non-negative eigenvalues.
2.2. Formulation of the Control Problem
The problem we address in this paper is to com-pute all the density matrices the system can reach for the quantum dissipative system which evolves under the Lindblad equation of motion given by eq. (1). We as-sume that we can apply any desired sequence of unitary transformations to the system, over a time scale of its coupling to the bath. For the purpose of this paper, we will confine our attention to two level systems, such as, e.g., a two level atom where the Hamiltonian H(t) is a time varying dipole term arising from a high bandwidth applied lase field. Our results will be largely generaliz-able to systems with arbitrary dimension.
3. Reformulation of the Problem in Terms
of the Spectrum of ρ
In this section we develop a general formalism that highlights the cooperative interplay between Hamilto-nian and dissipative dynamics. Following [27, 4], we assume that the action of the control Hamiltonian can be produced on a time scale fast compared with dissi-pation. We assume that the control Hamiltonian H(t) can produce any unitary transformation U ∈ SU (2) in the 2−level system, i.e. the system of interest is unitar-ily controllable. Combining these two assumptions we
have that any unitary transformation can be produced on the system in negligible time compared to the dissi-pation.
The above dynamical assumptions lead to another very important simplification. Since we have assumed that all unitary transformations in SU (2) can be pro-duced instantaneously, this includes bringing the den-sity matrix into diagonal form. As a result, the differ-ent elemdiffer-ents of each orbit can be considered redundant, and the orbit of ρ can be completely represented by its diagonal form, or ’spectrum’, λ (ρ). This suggests re-formulating the control problem entirely in terms of the spectrum, rather than in terms of ρ itself. The key step in this reformulation is to replace the equation of mo-tion for ρ, eq. (1), with an equamo-tion of momo-tion for the spectrum. We do this in the next section. The controls will enter into the equation in a modified way that gives additional insight into the interplay of Hamiltonian and dissipative dynamics.
Let Λ be its associated diagonal form of density matrix ρ.
Substitute ρ(t) = U (t)Λ(t)U†(t) into Eq.(1), we get
˙
ρ (t) = ˙U(t)Λ(t)U†(t) +U (t) ˙Λ(t)U†(t) +U (t)Λ(t) ˙U†(t)
= − iH0(t)U (t)Λ(t)U†(t) +U (t) ˙Λ(t)U†(t) +U (t)Λ(t)U†(t)iH0(t)
= − i[H0(t),U (t)Λ(t)U†(t)] +U (t) ˙Λ(t)U†(t) = − i[H(t),U (t)Λ(t)U†(t)] + L[U (t)Λ(t)U†(t)]
(2)
where H0(t) is defined by ˙U(t) = −iH0(t)U (t). We ob-tain
˙
Λ(t) =U†(t){−i[H(t) − H0(t),U (t)Λ(t)U†(t)] + L[U (t)Λ(t)U†(t)]}U (t)
= − i[U†(t)(H(t) − H0(t))U (t), Λ(t)] +U†(t)L[U (t)Λ(t)U†(t)]U (t)
(3)
Note that the left side of the above equation is a diagonal matrix, so for the right side we only need to keep the diagonal part. It is easy to see that the diagonal part is zero for the first term, thus we get
˙
Λ(t) = diag(U†(t)L[U (t)Λ(t)U †(t)]U (t)) (4) where we use diag(A) denote a diagonal matrix whose diagonal entries are the same as matrix A.
4. Reachable set for single spin
4.1. Examples
Let’s first work out two examples. We first study the single spin with pure decoherence in the z-basis. In this case,
L(ρ) = −γ[σz, [σz, ρ]]
So ˙
Λ(t) =diag(U†L(U ΛU†)U )
=diag(−U†γ [σz, [σz,U ΛU†]]U )
=diag(−γ[U†σzU, [U†σzU, Λ]]) (5) We can write Λ(t) =1 2I+ λ (t)σz where λ (t) ∈ [0,12], and U†(t)σzU(t) = a1(t)σx+ a2(t)σy+ a3(t)σz
where ∑3i=1a2i(t) = 1. Substitute these into above
equa-tions, we get ˙ λ (t) = −4γ (a21(t) + a22(t))λ (6) so λ (T ) = exp[ Z T 0 −4γ(a 2 1(t) + a22(t))dt]λ (0).
Note that the quantity a21(t) + a22(t) ∈ [0, 1], and by choosing appropriate U (t), it can be any value in this interval. So at time T , λ (T ) can be any value in [exp(−4γT )λ (0), λ (0)], i.e., the reachable set for the single spin in this case is
ρ (T ) = {U (1
2I+ λ (T )σz)U
†|
λ (T ) ∈ [exp(−4γ T )λ (0), λ (0)],U ∈ SU (2)} (7)
Let’s look at another example with both longitudi-nal and transverse relaxation.
L(ρ) = −γ1[σz, [σz, ρ]] − γ2[σx, [σx, ρ]]
In this case ˙
Λ(t) =diag(U†L(U ΛU†)U )
=diag(−U†γ1[σz, [σz,U ΛU†]]U
−U† γ2[σx, [σx,U ΛU†]]U ) =diag(−γ1[U†σzU, [U†σzU, Λ]] − γ2[U†σxU, [U†σxU, Λ]]) (8) WeC11.5 2499
Again we write Λ(t) =1 2I+ λ (t)σz where λ (t) ∈ [0,12], and U†(t)σzU(t) = a1(t)σx+ a2(t)σy+ a3(t)σz U†(t)σxU(t) = b1(t)σx+ b2(t)σy+ b3(t)σz
where ∑3i=1a2i(t) = 1, ∑3i=1b2i(t) = 1 and ∑3i=1aibi= 0.
Substitute these into the equation, we get ˙ λ (t) = [−4γ1(a21(t) + a22(t)) − 4γ2(b21(t) + b22(t))]λ (9) As a3b3= −a1b1− a2b2, (a3b3)2= (a1b1+ a2b2)2≤ (a21+ a22)(b21+ b22) = (1 − a23)(1 − b23) = 1 − a23− b2 3+ a23b23 (10) We get a23+ b23≤ 1, so 1 ≤ a21+ a22+ b21+ b22≤ 2 Assume γ1≥ γ2, then 4γ1(a21(t)+a22(t))+4γ2(b21(t)+b22(t)) ∈ [4γ2, 4(γ1+γ2)]
From this, it is easy to see that at time T ,
λ (T ) ∈ [exp(−4(γ1+ γ2)T )λ (0), exp(−4γ2T)λ (0)],
so the reachable set for the single spin in this case is ρ (T ) = {U (1
2I+ λ (T )σz)U
†|λ (T ) ∈
[exp(−4(γ1+ γ2)T )λ (0), exp(−4γ2T)λ (0)],U ∈ SU(2)}
4.2. General case
The examples above are just two special cases of the following general result.
Take the general expression of the master equation ˙ ρ = −i[H, ρ ] + L(ρ ) where L(ρ) =
∑
α β aα β(Fαρ F † β− 1 2{F † βFα, ρ})For the single spin, we can take the basis {Fα} as
nor-malized Pauli spin operators √1
2{σx, σy, σz}. The coef-ficient matrix A= axx axy axz ayx ayy ayz azx azy azz
known as the GKS(Gorini, Kossakowski and Sudar-shan) matrix [33], is semi-positive definite. For the mo-ment assume that the Markovian quantum dynamics is unital, which just means L(I) = 0. In the single spin case, this is equivalent to the condition that all the en-tries of the GKS matrix are real numbers[40].
˙
Λ(t) = diag(U†L(U ΛU†)U ) = diag(
∑
α β aα β(U †F αU ΛU†F † βU −1 2{U †F† βUU †F αU, Λ})) = diag(∑
α β aα β(U †F αU ΛU †F βU −1 2{U †F βUU †F αU, Λ})) (11)For the last step we just used the fact that Fβ is a Pauli
matrix which is Hermitian. Now U†FαU= cα γFγ where C = cxx cxy cxz cyx cyy cyz czx czy czz
∈ SO(3) is the adjoint representation of U . Substituting these expressions into equation(11), we obtain ˙ Λ(t) = diag(
∑
α β a0α β(FαΛFβ− 1 2{FβFα, Λ})), (12) where a0α β= cγ αaγ µcµ βis the transformed GKS matrix, i.e., A0= CTAC
Substituting Λ(t) = 12I+ λ (t)σz into equation (12), we
obtain the dynamics for λ (t), ˙
λ (t) = −(a0xx+ a0yy)λ (t)
Using Schur and Horn’s theorem on majorization (see appendix), we obtain
µ3+ µ2≤ a0xx+ a 0
yy≤ µ2+ µ1
where µ1≥ µ2≥ µ3are eigenvalues of the GKS matrix.
From this it is easy to see that at time T , all the values λ (T ) can be are
So the reachable set for the single spin under unital mas-ter quantum dynamics is
ρ (T ) = {U 1 2+ λ (T ) 0 0 12− λ (T ) U†|λ (T ) ∈ [e−(µ1+µ2)Tλ (0), e−(µ2+µ3)Tλ (0)],U ∈ SU (2)}
Important examples of unital master equation in-cludes phase damping and depolarizing, whose GKS matrix is just γI(I is the identity matrix). But there is another important decoherence mechanism which is not unital—amplitude damping, which models the sponta-neous emission from |1i to |0i, we will study it in the next section.
5. Two level dissipative system
The Lindblad form for two level system with spon-taneous emission from |1i to |0i is given by
L(ρ) = 2γσ−ρ σ+− γ{σ+σ−, ρ} = γ([σ−ρ , σ+] + [σ−, ρσ+]) (13) where σ+= σx+ iσy= 0 2 0 0 (14) σ−= σx− iσy= 0 0 2 0 (15) Similarly we write Λ(t) =12I+ λ (t)σz, 0 ≤ λ (t) ≤ 1 2, and U†(t)σxU(t) = a1(t)σx+ a2(t)σy+ a3(t)σz U†(t)σyU(t) = b1(t)σx+ b2(t)σy+ b3(t)σz
where ∑3i=1a2i(t) = 1, ∑3i=1b2i(t) = 1 and ∑3i=1aibi= 0.
Substituting these into the equation ˙
Λ(t) = diag(U†L(U ΛU†)U ), we obtain
˙
λ (t) =γ [−4(a21(t) + a22(t) + b21(t) + b22(t))λ + 4(a2(t)b1(t) − a1(t)b2(t))]
(16)
It is convenient to study the dynamics of 12+ λ (t):
d dt( 1 2+ λ (t)) = γ{−4[a 2 1(t) + a22(t) + b21(t) + b22(t)]× (1 2+ λ (t)) + 2[a2(t) + b1(t)] 2+ 2[a 1(t) − b2(t)]2}
Let’s look at the right side of the above equation to see what value it can take. As we have shown before,
1 ≤ a21(t) + a22(t) + b21(t) + b22(t) ≤ 2 So −8[1 2+ λ (t)] ≤ − 4(a 2 1(t) + a22(t) + b21(t) + b22(t))× (1 2+ λ (t)) ≤ − 4[1 2+ λ (t)] (17)
And 2[a2(t) + b1(t)]2+ 2[a1(t) − b2(t)]2≥ 0, so
γ [−4(a21(t) + a22(t) + b21(t) + b22(t))( 1 2+ λ (t)) + 2[a2(t) + b1(t)]2+ 2[a1(t) − b2(t)]2] ≥ −8γ[ 1 2+ λ (t)] and the equality is achievable at a1= b2= 1, a2= b1=
0, which gives the lower bound of dtd(12+ λ (t)). Now let’s find the upper-bound. Suppose
a21(t) + a22(t) + b21(t) + b22(t) = c where c ∈ [1, 2], as (a2+ b1)2+ (a1− b2)2≤2(a21(t) + a22(t) + b21(t) + b22(t)) =2c We get γ {−4(a21(t) + a22(t) + b21(t) + b22(t))× [1 2+ λ (t)] + 2(a2+ b1) 2+ 2(a 1− b2)2} ≤ γ{−4c[1 2+ λ (t)] + 4c} = 4γc[1 2− λ (t)] ≤ 8γ[1 2− λ (t)] (18)
The equality is achievable at a2= b1= 1, a1= b2= 0,
which gives the upper bound.
By connecting property, we know that dtd(12+ λ (t)) can take any value in
[−8γ(1
2+ λ (t)), 8γ( 1
2− λ (t))] With this, we can find the reachable set for λ (T ).
First for the lower bound, we always take the right side of eq.( 17) to be the smallest:
d dt[ 1 2+ λ (t)] = −8γ[ 1 2+ λ (t)] WeC11.5 2501
So 1 2+ λ (T ) = exp(−8γT )[ 1 2+ λ (0)] λ (T ) = exp(−8γ T )[1 2+ λ (0)] − 1 2 Similarly for the upper bound:
d dt[ 1 2+ λ (t)] = 8γ[ 1 2− λ (t)] = −8γ[1 2+ λ (t)] + 8γ (19)
The solution in this case is 1 2+ λ (T ) = exp(−8γT )( 1 2+ λ (0)) + exp(−8γT ) Z T 0 8γ exp(8γt)dt = exp(−8γT )(1 2+ λ (0)) + 1 − exp(−8γT ) (20) So λ (T ) = exp(−8γ T )(−1 2+ λ (0)) + 1 2 Thus the reachable set for this system is
ρ (T ) = {U [1 2I+ λ (T )σz]U †|U ∈ SU(2), λ (T ) ∈ [max{0, exp(−8γ T )[1 2+ λ (0)] − 1 2}, exp(−8γT )(−1 2+ λ (0)) + 1 2]} (21)
The intuitive interpretation of this is as follows. For times short compared with 1γ, the reachable spectra are close to the original spectrum. For longer times, how-ever, one can play the tendency of the system to relax off against the ability to perform unitary control to ma-nipulate the spectrum in any desired faction, so that for T >>1
γ, essentially all possible states can be reached.
6. Conclusion
Control of open quantum systems is an impor-tant problem for a wide variety of physics, chemistry, and engineering applications. This paper analyzed the problem of controlling open quantum systems in cases where full, high-bandwidth coherent control of the sys-tem is available. This coherent control can be used to ‘present’ various aspects of the system’s state to the en-vironmental interaction. Because of the presence of fast coherent control, the quantity of interest under control is the spectrum of the density matrix. We analyzed the reachability of various spectral forms for two-level sys-tems and derived general formulae for reachability in
the presence of pure decoherence, and of decoherence and relaxation. In the presence of pure decoherence, the time evolution is unital, and tends inevitably towards the fully mixed state. Coherent control can only de-lay this process and achieve a variety of less than fully mixed states at various times along the way. The pres-ence of relaxation in addition to decoherpres-ence allows a richer set of states to attained by playing decoherence (which drives the system to a fully mixed state ) and relaxation (which drives the system to a pure state) off against each other.
A. Majorization
For an element x = (x1, ..., xk)Tof Rkwe denote by
x↓= (x↓1, ..., x↓k)T a permutation of x so that x↓ i ≥ x
↓ j if
i< j, where 1 ≤ i, j ≤ k.
Definition 1 (majorization) A vector x ∈ Rk is
ma-jorized by a vector y ∈ Rk(denoted x ≺ y), if d
∑
j=1 x↓j≤ d∑
j=1 y↓j (22)for d = 1, . . . , k − 1, and the inequality holds with equal-ity when d = k.
Proposition 1 (Schur, Horn [39, 41]) For an element λ = (λ1, ..., λn)T, let Dλ be a diagonal matrix with
(λ1, ..., λn) as its diagonal entries, let a = (a1, ..., an)T
be the diagonal entries of matrix A = KTDλK, where
K∈ SO(n). Then a ≺ λ . Conversely for any vector a≺ λ , there exists a K ∈ SO(n), such that (a1, ..., an)T
are the diagonal entries of A = KTD λK
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