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.
As Published
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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Thu May 26 08:55:42 EDT 2016
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http://hdl.handle.net/1721.1/74601
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Detailed Terms
Joint 48th IEEE Conference on Decision and Control and
28th Chinese Control Conference
Shanghai, P.R. China, December 16-18, 2009
WeC11.5
Control of single spin in Markovian environment
Haidong Yuan∗
Abstract
equation, which takes the form
ρ̇ = −i[H(t), ρ] + L(ρ)
In this article we study the control of single spin in
Markovian environment. Given an initial state, we compute 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)
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],
1
L(ρ) = ∑ aαβ (Fα ρFβ† − {Fβ† Fα , ρ}),
2
ij
1. Introduction
where Fα , Fβ are the Lindblad operators. Eq. (1) has
the following three well known properties: 1) Tr(ρ) remains unity for all time, 2) ρ remains a Hermitian matrix, and 3) ρ stays positive semi-positive definite, i.e.
that ρ never develops non-negative eigenvalues.
In the last two decades, control theory has been applied to an increasingly wide number of problems in
physics and chemistry whose dynamics are governed
by the time-dependent Schrödinger 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 computing [19, 20, 21], oriented rotational wavepackets
[22], etc. [23, 24]. More recently, there has been vigorous effort in studying the control of open quantum systems 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 Lindblad equation is an extension of the TDSE that allows
for the inclusion of dissipative processes. In this article, we study two level systems governed by the controlled Lindblad equation, we will compute all the possible states to which the systems can be driven at arbitrary time, under the assumption that fast unitary operations on the system are available.
2.2. Formulation of the Control Problem
The problem we address in this paper is to compute 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 assume 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 generalizable to systems with arbitrary dimension.
3. Reformulation of the Problem in Terms
of the Spectrum of ρ
2. Setting up the control problem
In this section we develop a general formalism that
highlights the cooperative interplay between Hamiltonian 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 dissipation. 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 unitarily controllable. Combining these two assumptions we
2.1. The system equations of motion and the
Lindblad formula for dissipation
Let ρ denote the density matrix of an quantum system. The density matrix evolves under the Lindblad
∗ Haidong Yuan is with the department of Mechanical Engineering,
Massachusetts Institute of Technology.
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
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WeC11.5
4. Reachable set for single spin
have that any unitary transformation can be produced
on the system in negligible time compared to the dissipation.
The above dynamical assumptions lead to another
very important simplification. Since we have assumed
that all unitary transformations in SU(2) can be produced instantaneously, this includes bringing the density matrix into diagonal form. As a result, the different elements 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 reformulating 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 motion for ρ, eq. (1), with an equation of motion 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
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
Λ̇(t) =diag(U † L(UΛU † )U)
=diag(−U † γ[σz , [σz ,UΛU † ]]U)
†
=diag(−γ[U σzU, [U σzU, Λ]])
We can write
1
Λ(t) = I + λ (t)σz
2
where λ (t) ∈ [0, 21 ], and
U † (t)σzU(t) = a1 (t)σx + a2 (t)σy + a3 (t)σz
where ∑3i=1 a2i (t) = 1. Substitute these into above equations, we get
ρ̇(t) =U̇(t)Λ(t)U † (t) +U(t)Λ̇(t)U † (t)
λ̇ (t) = −4γ(a21 (t) + a22 (t))λ
†
+U(t)Λ(t)U̇ (t)
(2)
+U(t)Λ(t)U † (t)iH 0 (t)
−4γ(a21 (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
= − i[H(t),U(t)Λ(t)U † (t)] + L[U(t)Λ(t)U † (t)]
where H 0 (t) is defined by U̇(t) = −iH 0 (t)U(t). We obtain
1
ρ(T ) = {U( I + λ (T )σz )U † |
2
(7)
λ (T ) ∈ [exp(−4γT )λ (0), λ (0)],U ∈ SU(2)}
Λ̇(t) =U † (t){−i[H(t) − H 0 (t),U(t)Λ(t)U † (t)]
(3)
Let’s look at another example with both longitudinal and transverse relaxation.
+U † (t)L[U(t)Λ(t)U † (t)]U(t)
L(ρ) = −γ1 [σz , [σz , ρ]] − γ2 [σx , [σx , ρ]]
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
Λ̇(t) = diag(U † (t)L[U(t)Λ(t)U†(t)]U(t))
Z T
λ (T ) = exp[
0
= − i[H 0 (t),U(t)Λ(t)U † (t)] +U(t)Λ̇(t)U † (t)
= − i[U † (t)(H(t) − H 0 (t))U(t), Λ(t)]
(6)
so
= − iH 0 (t)U(t)Λ(t)U † (t) +U(t)Λ̇(t)U † (t)
+ L[U(t)Λ(t)U † (t)]}U(t)
(5)
†
In this case
Λ̇(t) =diag(U † L(UΛU † )U)
=diag(−U † γ1 [σz , [σz ,UΛU † ]]U
(4)
−U † γ2 [σx , [σx ,UΛU † ]]U)
†
†
=diag(−γ1 [U σzU, [U σzU, Λ]]
where we use diag(A) denote a diagonal matrix whose
diagonal entries are the same as matrix A.
− γ2 [U † σxU, [U † σxU, Λ]])
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(8)
WeC11.5
known as the GKS(Gorini, Kossakowski and Sudarshan) matrix [33], is semi-positive definite. For the moment 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 entries of the GKS matrix are real numbers[40].
Again we write
1
Λ(t) = I + λ (t)σz
2
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
Λ̇(t) = diag(U † L(UΛU † )U)
where ∑3i=1 a2i (t) = 1, ∑3i=1 b2i (t) = 1 and ∑3i=1 ai bi = 0.
Substitute these into the equation, we get
= diag(∑aαβ (U † Fα UΛU † Fβ†U
λ̇ (t) = [−4γ1 (a21 (t) + a22 (t)) − 4γ2 (b21 (t) + b22 (t))]λ (9)
1
− {U † Fβ†UU † Fα U, Λ}))
2
= diag(∑aαβ (U † Fα UΛU † Fβ U
αβ
As a3 b3 = −a1 b1 − a2 b2 ,
αβ
(a3 b3 )2 = (a1 b1 + a2 b2 )2 ≤ (a21 + a22 )(b21 + b22 )
= (1 − a23 )(1 − b23 )
(11)
1
− {U † Fβ UU † Fα U, Λ}))
2
(10)
= 1 − a23 − b23 + a23 b23
For the last step we just used the fact that Fβ is a Pauli
matrix which is Hermitian. Now
We get a23 + b23 ≤ 1, so
U † Fα U = cαγ Fγ
1 ≤ a21 + a22 + b21 + b22 ≤ 2


cxx cxy cxz
where C =  cyx cyy cyz  ∈ SO(3) is the adjoint
czx czy czz
representation of U. Substituting these expressions into
equation(11), we obtain
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(−4γ2 T )λ (0)],
so the reachable set for the single spin in this case is
1
Λ̇(t) = diag(∑ a0αβ (Fα ΛFβ − {Fβ Fα , Λ})),
2
αβ
1
ρ(T ) = {U( I + λ (T )σz )U † |λ (T ) ∈
2
[exp(−4(γ1 + γ2 )T )λ (0), exp(−4γ2 T )λ (0)],U ∈ SU(2)}
where
4.2. General case
is the transformed GKS matrix, i.e.,
(12)
a0αβ = cγα aγ µ cµβ
A0 = CT AC
The examples above are just two special cases of
the following general result.
Take the general expression of the master equation
Substituting Λ(t) = 12 I + λ (t)σz into equation (12), we
obtain the dynamics for λ (t),
ρ̇ = −i[H, ρ] + L(ρ)
λ̇ (t) = −(a0xx + a0yy )λ (t)
where
Using Schur and Horn’s theorem on majorization (see
appendix), we obtain
1
L(ρ) = ∑ aαβ (Fα ρFβ† − {Fβ† Fα , ρ})
2
αβ
µ3 + µ2 ≤ a0xx + a0yy ≤ µ2 + µ1
For the single spin, we can take the basis {Fα } as normalized Pauli spin operators √12 {σx , σy , σz }. The coefficient matrix


axx axy axz
A =  ayx ayy ayz 
azx azy azz
where µ1 ≥ µ2 ≥ µ3 are eigenvalues of the GKS matrix.
From this it is easy to see that at time T , all the values
λ (T ) can be are
[e−(µ1 +µ2 )T λ (0), e−(µ2 +µ3 )T λ (0)]
2500
WeC11.5
Let’s look at the right side of the above equation to
see what value it can take. As we have shown before,
So the reachable set for the single spin under unital master quantum dynamics is
1
0
2 + λ (T )
ρ(T ) = {U
U † |λ (T ) ∈
1
0
−
λ
(T
)
2
1 ≤ a21 (t) + a22 (t) + b21 (t) + b22 (t) ≤ 2
So
[e−(µ1 +µ2 )T λ (0), e−(µ2 +µ3 )T λ (0)],U ∈ SU(2)}
1
−8[ + λ (t)] ≤ − 4(a21 (t) + a22 (t) + b21 (t) + b22 (t))×
2
1
(17)
( + λ (t))
2
1
≤ − 4[ + λ (t)]
2
Important examples of unital master equation includes 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 spontaneous emission from |1i to |0i, we will study it in the
next section.
And 2[a2 (t) + b1 (t)]2 + 2[a1 (t) − b2 (t)]2 ≥ 0, so
5. Two level dissipative system
1
γ[−4(a21 (t) + a22 (t) + b21 (t) + b22 (t))( + λ (t))
2
1
+ 2[a2 (t) + b1 (t)]2 + 2[a1 (t) − b2 (t)]2 ] ≥ −8γ[ + λ (t)]
2
The Lindblad form for two level system with spontaneous emission from |1i to |0i is given by
L(ρ) = 2γσ− ρσ+ − γ{σ+ σ− , ρ}
= γ([σ− ρ, σ+ ] + [σ− , ρσ+ ])
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
(13)
a21 (t) + a22 (t) + b21 (t) + b22 (t) = c
where
where c ∈ [1, 2], as
σ+ = σx + iσy =
σ− = σx − iσy =
1
2,
0
0
2
0
0
2
0
0
(a2 + b1 )2 + (a1 − b2 )2 ≤2(a21 (t) + a22 (t) + b21 (t) + b22 (t))
(14)
=2c
(15)
We get
Similarly we write Λ(t) = 21 I + λ (t)σz , 0 ≤ λ (t) ≤
and
γ{−4(a21 (t) + a22 (t) + b21 (t) + b22 (t))×
1
[ + λ (t)] + 2(a2 + b1 )2 + 2(a1 − b2 )2 }
2
1
≤ γ{−4c[ + λ (t)] + 4c}
2
1
= 4γc[ − λ (t)]
2
1
≤ 8γ[ − λ (t)]
2
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=1 a2i (t) = 1, ∑3i=1 b2i (t) = 1 and ∑3i=1 ai bi = 0.
Substituting these into the equation
Λ̇(t) = diag(U † L(UΛU † )U),
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
we obtain
λ̇ (t) =γ[−4(a21 (t) + a22 (t) + b21 (t) + b22 (t))λ
+ 4(a2 (t)b1 (t) − a1 (t)b2 (t))]
(18)
(16)
1
1
[−8γ( + λ (t)), 8γ( − λ (t))]
2
2
It is convenient to study the dynamics of 12 + λ (t):
d 1
( + λ (t)) = γ{−4[a21 (t) + a22 (t) + b21 (t) + b22 (t)]×
dt 2
1
( + λ (t)) + 2[a2 (t) + b1 (t)]2 + 2[a1 (t) − b2 (t)]2 }
2
2501
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 1
1
[ + λ (t)] = −8γ[ + λ (t)]
dt 2
2
WeC11.5
So
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 delay this process and achieve a variety of less than fully
mixed states at various times along the way. The presence of relaxation in addition to decoherence 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.
1
1
+ λ (T ) = exp(−8γT )[ + λ (0)]
2
2
1
1
λ (T ) = exp(−8γT )[ + λ (0)] −
2
2
Similarly for the upper bound:
d 1
1
[ + λ (t)] = 8γ[ − λ (t)]
dt 2
2
1
= −8γ[ + λ (t)] + 8γ
2
(19)
The solution in this case is
A. Majorization
1
1
+ λ (T ) = exp(−8γT )( + λ (0))
2
2
Z T
+ exp(−8γT )
8γ exp(8γt)dt
(20)
0
1
= exp(−8γT )( + λ (0)) + 1 − exp(−8γT )
2
For an element x = (x1 , ..., xk )T of Rk we denote by
= (x1↓ , ..., xk↓ )T a permutation of x so that xi↓ ≥ x↓j if
i < j, where 1 ≤ i, j ≤ k.
x↓
Definition 1 (majorization) A vector x ∈ Rk is majorized by a vector y ∈ Rk (denoted x ≺ y), if
So
1
1
λ (T ) = exp(−8γT )(− + λ (0)) +
2
2
Thus the reachable set for this system is
d
∑ x↓j ≤
j=1
1
ρ(T ) = {U[ I + λ (T )σz ]U † |U ∈ SU(2),
2
1
1
λ (T ) ∈ [max{0, exp(−8γT )[ + λ (0)] − }, (21)
2
2
1
1
exp(−8γT )(− + λ (0)) + ]}
2
2
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, however, one can play the tendency of the system to relax
off against the ability to perform unitary control to manipulate the spectrum in any desired faction, so that for
T >> 1γ , essentially all possible states can be reached.
d
∑ y↓j
(22)
j=1
for d = 1, . . . , k − 1, and the inequality holds with equality 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 = K T Dλ K, 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 = K T Dλ K
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