Suppression of Decoherence of a Spin-Boson System by Time-Periodic Control

0.12 MB·

by  Volker Bach

Suppression of Decoherence
of a Spin-Boson System
by Time-Periodic Control
7
1
0
2 Volker Bach <[email protected]>,

n Alexander Hach <[email protected]>,
a
J
Institut fu¨r Analysis und Algebra
9
1 TU Braunschweig

] Pockelsstr. 14
h
p 38106 Braunschweig
-
h Germany
t
a
m
19-Jan-2017
[

1
v
6 Abstract
3
4 We consider a finite-dimensional quantum system coupled to the bosonic
5
radiation field and subject to a time-periodic control operator. Assuming
0
. the validity of a certain dynamic decoupling condition we approximate the
1
0 system’s time evolution with respect to the non-interacting dynamics. For
7 sufficientlysmallcouplingconstantsgandcontrolperiodsT weshowthata
1
certaindeviationofcoupledanduncoupledpropagatormaybeestimatedby
:
v O(gtT). Our approach relies on the concept of Kato stability and general
i
X theoryonnon-autonomous linearevolution equations.
r
a
Keywords: Decoherence · Quantum control theory · Open quantum systems ·
Kato stability
SuppressionofDecoherence-19-Jan-2017 1
1 Result and Discussion
We consider an open quantum system consisting of a small, finite-dimensional
systemScoupledto areservoirR withinfinitelymanydegrees offreedom.
Specifically, we assume the small system to be an N-level atom, for some
N ≥ 2, i.e., the system’s Hilbert space is H = CN, with a dynamics generated
S
by aself-adjointHamiltonianmatrix
H = diag E ,E ,...,E ,E , (1)
S N−1 N−2 1 0
(cid:0) (cid:1)
which we assume to be diagonal with nonnegative, nondegenerate eigenvalues
E > E > ... > E > E ≥ 0.
N−1 N−2 1 0
The reservoir Hilbert space F = F [h] is the boson Fock space over the
R b
square-integrable functions h := L2(R3) on R3 and carries a three-dimensional,
masslessscalarquantumfield–acaricatureofthephotonfield–whosedynamics
is generated bythesecond quantization
H := dΓ(ω) = ω(k)a∗a d3k (2)
R k k
ZR3
of (the operator of multiplication by) the photon dispersion ω(k) := |k|. Here,
{a ,a∗} defines the standard Fock representation of the canonical commuta-
k k k∈R3
tionrelation (CCR)
[a , a ] = [a∗ , a∗] = 0, [a ,a∗] = δ(p−k), a Ω = 0, (3)
p k p k p k k
for all k,p ∈ R3, as an operator-valued distribution, with Ω ∈ F being the
R
normalizedvacuumvector.
The Hilbert space of the composite atom-photon system S + R is the tensor
product space H = H ⊗F . Without interaction between these two compo-
SR S R
nents,thedynamicsisgenerated by theself-adjointHamiltonian
H(0) := H +H , (4)
SR S R
where here and henceforth we leave out trivial tensor factors whenever possible
and identity,e.g., H ≡ H ⊗1 andH ≡ 1 ⊗H .
S S R R S R
Adipole-typeinteractiongH couplestheN-levelatomto thelargereservoir,
I
i.e., thefull,interactingdynamicsisgenerated by theself-adjointHamiltonian
H(g) := H(0) +gH . (5)
SR SR I
Here, g > 0 isasmallcouplingconstantand
H := Q⊗φ(f) ≡ Qφ(f) = Q a∗(f)+a(f) (6)
I
(cid:0) (cid:1)
SuppressionofDecoherence-19-Jan-2017 2
istheself-adjointinteractionoperatorspecifiedbyaself-adjointcomplexN×N-
matrixQ = Q∗ timesthefield operatorφ(f). Furthermore, forf ∈ h,
a∗(f) := f(k) a∗ d3k, a(f) := f(k) a d3k. (7)
k k
Z Z
We assume that f,ω−1f ∈ h which implies the semiboundedness and self-
adjointness of H(g) on the domain of H(0), for any g > 0, since under this as-
SR SR
sumptionH is an infinitesimalperturbationofH(0).
I SR
Thanks to the self-adjointness of H(g), the evolution operator it generates ac-
SR
cording to Schro¨dinger, i.e., the solution of the initial value problem ∂ U(g)(t) =
t SR
−iH(g)U(g)(t), U(g)(0) = 1, is the strongly continuous one-parameter unitary
SR SR SR
group t 7→ exp[−itH(g)]. Given an initial state of the atom-photon system by a
SR
densitymatrixρ ∈ L1(H),i.e., apositiveoperatorofunittrace, thestateattime
0 +
t ∈ R is givenby
ρ = exp[−itH(g)] ρ exp[itH(g)]. (8)
t SR 0 SR
Any initial state ρ eventually evolves into the ground state or the thermal equi-
0
librium state at zero or positive temperature, respectively, as time t → ∞ grows
large. This phenomenon is usually refered to as return to equilibrium. As a con-
sequence, after a sufficiently long time has elapsed, the state becomes incoherent
and any information initiallyencoded in it is lost. A quantum computer can only
process data reliably if its calculations are finished long before the loss of coher-
ence due to the dissipative process of return to equilibrium described above sets
in.
Further perturbationsadditionallyacting on the systemwould typicallyspeed
up thedecoherence process. Iftheperturbationis suitablydesigned,however,the
opposite effect might occur and decoherence is suppressed by the perturbation,
rather thanenhanced.
The present paper is devoted to thequestion under which conditionsthis sup-
pression of decoherence occurs. More specifically, we study the influence of a
time-periodicperturbationrepresented by acontroloperator H (t) which acts on
C
the small system S only. This latter restriction is a minimal requirement for a
physically realistic model: H (t) cannot change the environment. The control
C
operator is assumed to be a continuous family H ∈ C[R;B(H )] of self-adjoint
C S
complex N × N matrices such that H (t + T) = H (t), for some time period
C C
T > 0 and allt ∈ R.
Acting on the small system as an external force, the generator of the full dy-
namics includingthecontrol operatorH (t) is
C
H(g) (t) := H(g) +H (t) = H +H +H (t)+gH . (9)
SRC SR C S R C I
SuppressionofDecoherence-19-Jan-2017 3
Thetheoryofnon-autonomouslinearevolutionequationsensuresthatforthecor-
respondingtime-dependentSchro¨dingerequation
∂ U(g) (t,s) = −iH(g) (t)U(g) (t,s), U(g) (s,s) = 1,
t SRC SRC SRC SRC
 (10)
∂ U(g) (t,s) = iU(g) (t,s)H(g) (s), U(g) (t,t) = 1,
s SRC SRC SRC SRC
thereexistsauniquefamilyU(g) ∈ C1[∆;B(H )]ofunitaryoperators onH ,
SRC SR SR
where ∆ := {(s,t) ∈ R2|s ≤ t} ⊆ R2, thatsolves(10).
Our main result is Theorem 1 below which asserts that, under Decoupling
Condition (14), the deviation of U(g) (t,0) from the identity is of order O(gtT),
SRC
fortimessmallerthang−1. Thisistobecomparedtothedeviationofexp[−itH(g)]
SR
from the identity which is of order O(gt). So, for sufficiently small time periods
T > 0, the control operator effectively slows down the evolution and hence also
thedecoherence ofthesystem.
Toformulatethedecouplingcondition,wedenotebyU ∈ C1[∆;B(H )]the
C S
propagatorgenerated byH (t), i.e.,theuniquesolutionof
C
∂ U (t,s) = −iH (t)U (t,s), U (s,s) = 1,
t C C C C
 (11)
∂ U (t,s) = iU (t,s)H (s), U (t,t) = 1,
s C C C C
and Q(τ) :=U (τ,0)QU (τ,0)∗ on H . Ourmainresult isas follows.
C C S
Theorem 1. Let L ∈ N, assumethatω−1f ∈ h and ωL+2f ∈ h,and set
e
L+2
L+2
M := 2 (ω−1/2 +1)f + 2 (ωk +1)f , (12)
2 (cid:18) k (cid:19) 2
(cid:13) (cid:13) Xk=1 (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
C(0) := 1 + kH k + sup kH (r)k. (13)
SRC S C
0≤r≤T
FurtherassumethatgkQkMT ≤ 1 andthatthefollowingdecoupling condition
T T
Q(τ)dτ = U (τ,0)QU (τ,0)∗ dτ = 0 (14)
C C
Z Z
0 0
e
holds true. Then, for any t ≥ 0 and with n := t , δ := t − n · T as well as
T
C := M kQkg ·max{1,4C(0) +3M kQkg},(cid:4) (cid:5)
g SRC
e
(H +1)L U(g) (t,0)−U(0) (t,0) (H +1)−L−2
R SRC SRC R
(cid:13) (cid:13)
(cid:0) (cid:1)
(cid:13) (cid:13)
(cid:13) (0) (cid:13)
≤ T ·M kQkg δ + 4C +3M kQkg nT exp[kQkM gt]
SRC
h (cid:16) (cid:17) i
≤ T ·C t exp[kQkM gt]. (15)
g
e
SuppressionofDecoherence-19-Jan-2017 4
WediscussTheorem1:
• The idea of suppression of decoherence by a periodic control goes back
to [5]. Theorem 1 was proven with mathematical rigor in [3], but under
strongerassumptionsand withconsiderablymoreinvolvedmethods:
- First, the reservoir in [3] was assumed to represent a fermion, rather
thanabosonfield.
- Secondly, the control operator H (t) was assumed to commute with
C
theHamiltonianH oftheatom,[H (t),H ] = 0,forall t ∈ R.
S C S
- A third difference is the framework of Liouvilleans as generators of
the dynamicsat nonzero temperatures which is considerably more in-
volvedon atechnical level.
- On the otherhand, theapproach in [3] yields control on thedynamics
for all times – large and small – and, in particular, allows to follow
the rate of convergence to the limiting state, as t → ∞. In contrast,
the methods used in the present paper give nontrivial estimates only
for times less than g−1, which is large compared to unity but small
compared tothevanHovetimescale∼ g−2.
• We observethat Decoupling Condition (14) and ∂ Q(t) = −i[H (t),Q(t)]
t C
imply
e e
T T t
−T Q(0) = dt Q(t)−Q(0) = −i dt ds[H (s),Q(s)].
C
Z Z Z
0 (cid:0) (cid:1) 0 0
e e e e (16)
SincekQ(s)k = kQk,foralls ∈ [0,T],thetriangleinequalityhenceyields
e T 1
kH (t)kdt ≥ . (17)
C
Z 2
0
This estimate shows that due to Decoupling Condition (14) the action the
controloperatorexcertsonthesysteminasinglecycleisatleastoftheorder
of unity with respect to natural units (ℏ = 1). Assuming a control period
T corresponding to a physically feasible time resolution of a hypothetical
control operatorH (t), e.g. a femtosecondregimeT ∼ 10−15s, theenergy
C
densityinSI-unitsofsuchadeviceactingonanatom-sizedquantumsystem
wouldbeabout1011J/m3.
In the following Section 2 we review some standard material on solutions of
linearnon-autonomousevolutionequationsonBanach spacesforwhichwefocus
SuppressionofDecoherence-19-Jan-2017 5
on the special case of unitary propagators for the time-dependent Schro¨dinger
equation. In order to apply this theory to the present model situation of a spin-
boson model with a time-periodic control, we then derive the necessary relative
operator bounds. After these preparations, we proceed to the proof of Theorem 1
giveninSection 3.
SuppressionofDecoherence-19-Jan-2017 6
2 Propagators and Kato Stability
Inthissectionwerecallastandardsetofsufficientconditionsfortheexistenceofa
(unitary)propagator U(t,s) forthetime-dependentSchro¨dingerequation
(t,s)∈∆
(cid:0) (cid:1)
∂ U(t,s) = −iH(t)U(t,s), U(s,s) = 1,
t
∀(t,s) ∈ ∆ :  (18)
∂ U(t,s) = iU(t,s)H(t), U(t,t) = 1,
s

givenbyusingtheconceptofKato quasi-stability.
To define this notion we assume X,k · k to be a complex Banach space
with a dense Banach subspace Y ⊆(cid:0)X whos(cid:1)e norm k · k can be written as
Y
kxk = kΘxk for a suitable linear, isometric bijection Θ : Y → X. We further
Y
assumethatkΘxk ≥ kxk,forallx ∈ X. TheoperatorΘallowsustoavoidusing
b b
thenormk·k altogether.
bY b
Definition2. Let X,|·| beacomplexBanachspaceandY ⊆ X adenseBanach
subspace. A famil(cid:0)y G ≡(cid:1) G(t) of densely defined, closed operators G(t) is
t∈R+
0
called Kato quasi-stable,(cid:0)if the(cid:1)re exists a constant C ≥ 1 and continuous maps
β ,β : R+ → R+ suchthatfollowingconditionsB1,B2, andB3 aresatisfied:
0 1 0 0
B1 The operators G define a norm-continuous family of bounded operators
fromY toX, i.e.,GΘ−1 ∈ C R+,B(X) .
0
(cid:2) (cid:3)
B2 Thecommutators[Θb,G(t)]Θ−1 := ΘG(t)Θ−1−G(t)aredenselydefinedon
X andextendtoacontinuousfamilyofboundedoperators,[Θ,G(t)]Θ−1 ∈
b b b b
C R+,B(X) , with [Θ,G(t)]Θ−1 = β (t).
0 B(X) 1 b b
(cid:2) (cid:3) (cid:13) (cid:13)
B3 For all n ∈ N, all t(cid:13),.b..,t ∈bR+,(cid:13)and all λ > β (t ),...,λ > β (t ),
1 n 0 1 0 1 n 0 n
thenormestimate
n n
1
−1
λ −G(t ) ≤ C · (19)
k k
(cid:12) (cid:12) λ −β (t )
(cid:12)kY=1(cid:0) (cid:1) (cid:12) Yk=1 k 0 k
(cid:12) (cid:12)
(cid:12) (cid:12)
holdstrue.
One of the main results of the theory on non-autonomous linear evolution
equations is Theorem 3, below; see, e.g., [4, 1, 2]. A key element in the proof of
Theorem 3 in [1] and in [2] is the Yosida approximation G (t) := −λ +λ2[λ −
λ
G(t)]−1,forλ > β (t),whichdefines afamilyofboundedoperatorsthatstrongly
0
convergeto G(t),as λ → ∞.
SuppressionofDecoherence-19-Jan-2017 7
Theorem 3. Let X,| · | be a complex Banach space, Y ⊆ X a dense Banach
subspace, and G(cid:0)≡ G((cid:1)t) a Kato quasi-stable family of densely defined,
t∈R+
0
closed operators, wit(cid:0)h M (cid:1)≥ 1, β ,β : R+ → R+ corresponding to Condi-
0 1 0 0
tions B1, B2, and B3. Then there exists a unique solution U(t,s) for the
(t,s)∈∆
non-autonomouslinearevolutionequation (cid:0) (cid:1)
∂ U(t,s) = G(t)U(t,s), U(s,s) = 1,
t
∀(t,s) ∈ ∆ :  (20)
∂ U(t,s) = −U(t,s)G(t), U(t,t) = 1,
s

which obeys thefollowingnormbounds,
t
kU(t,s)k ≤ C β (τ)dτ, (21)
0
Z
s
t
ΘU(t,s)Θ−1 ≤ C β (τ)+Cβ (τ) dτ, (22)
0 1
Z
(cid:13) (cid:13) s (cid:8) (cid:9)
(cid:13)b b (cid:13)
for all(t,s) ∈ ∆.
If X is specified to be a complex Hilbert space H, D = Ran(Θ) ⊆ H, for
some unbounded, self-adjoint operator Θ ≥ 1, and G is a strongly continuous
b
family−iH ≡ −iH(t) ofskew-adjointoperatorsonH,thenConditionB3
t∈R+ b
0
in Definition 2(cid:0)is autom(cid:1)atic with C = 1 and β ≡ 0, and Theorem 3 can be
0
strengthenedto thefollowingassertion.
Theorem 4. Let H be a separable complex Hilbert space, D = Ran(Θ) ⊆ H,
for some unbounded, self-adjoint operator Θ ≥ 1, and H ≡ H(t) a
b t∈R+
0
strongly continuous family of self-adjoint operators H(t) = H∗((cid:0)t) on (cid:1)H such
b
that HΘ−1,[Θ,H(t)]Θ−1 ∈ C R+,B(H) . Then there exists a unique propaga-
0
tor U(t,s) tothetime-d(cid:2)ependentS(cid:3)chro¨dingerequation
b (bt,s)∈∆ b
(cid:0) (cid:1)
∂ U(t,s) = −iH(t)U(t,s), U(s,s) = 1,
t
∀(t,s) ∈ ∆ :  (23)
∂ U(t,s) = iU(t,s)H(t), U(t,t) = 1,
s

which isa familyofunitaryoperatorsfulfillingthenormestimate
t
ΘU(t,s)Θ−1 ≤ [Θ,G(τ)]Θ−1 dτ, (24)
Z
(cid:13) (cid:13) s (cid:13) (cid:13)
(cid:13)b b (cid:13) (cid:13) b b (cid:13)
for all(t,s) ∈ ∆.
SuppressionofDecoherence-19-Jan-2017 8
Toapply Theorem4 tothepresent modelsituation,wechoose
H := H , Θ := ΘL+2, Θ := H +1, H(t) := H (t). (25)
SR R SRC
b
To validatethehypothesisofTheorem 4, wedefine
M :=2 (ω−1/2 +1)f , (26)
−1/2 2
(cid:13) (cid:13)
(cid:13)n (cid:13)
n
M :=2 (ωk +1)f (27)
n (cid:18)k(cid:19) 2
Xk=1 (cid:13) (cid:13)
(cid:13) (cid:13)
and establishthefollowingbounds.
Lemma 5. Let n ∈ N andassumethatω−1/2f,ωnf ∈ L2(R3). Then
1
ka(f)Θ−1k, ka∗(f)Θ−1k ≤ M . (28)
−1/2
2
1
[Θn,a∗(f)]Θ−n , [Θn,a(f)]Θ−n ≤ M . (29)
n
2
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
Proof. ItisconvenienttointroducethesubspaceFfin ⊆ F offinitevectorswhose
R R
elements have only finitely many non-vanishing components, each being smooth
andcompactlysupported. Foranynormalizedfinitevectorψ ∈ Ffin,wehavethat
R
ka∗(ωkf)ψk2 = kωkfk2 + ka(ωkf)ψk2, (30)
2
forall k ≥ 0. Additionallyrequiringthat k ≥ 1, wefurtherhave
ka(ωkf)ψk ≤ |f(ξ)|kωk(ξ)a ψkdξ ≤ kfk · ψ dΓ[ω2k]ψ 1/2 (31)
ξ 2
Z
(cid:10) (cid:12) (cid:11)
(cid:12)
≤ kfk · ψ dΓ[ω] 2kψ 1/2 = kfk ·kHkψk ≤ kfk ·kΘkψk,
2 2 R 2
(cid:10) (cid:12)(cid:0) (cid:1) (cid:11)
(cid:12)
where dΓ(A) denotes the second quantization of an operator A. For k = 0, we
slightlymodifythisestimateand obtain
ka(f)ψk ≤ |f(ξ)|ka ψkdξ ≤ ω−1/2f ·kH1/2ψk. (32)
Z ξ 2 R
(cid:13) (cid:13)
(cid:13) (cid:13)
Thisestimateand (30)withk = 0 establish
ka(f)Θ−1k, ka∗(f)Θ−1k ≤ (ω−1/2 +1)f (33)
2
(cid:13) (cid:13)
(cid:13) (cid:13)
and hence(28).
SuppressionofDecoherence-19-Jan-2017 9
Ontheotherhand,Eq. (30)and (31)implyfork ≥ 1that
ka(ωkf)Θ−kk, ka∗(ωkf)Θ−kk ≤ (ωk +1)f . (34)
2
(cid:13) (cid:13)
Usingtheidentities (cid:13) (cid:13)
H ,a∗(f) = a∗(ωf), H ,a(f) = −a(ωf), (35)
R R
(cid:2) (cid:3) (cid:2) (cid:3)
and an induction,weeasily find that
Θna∗(f)Θ−n = Θn−1a∗(f)Θ−(n−1) + Θn−1a∗(ωf)Θ−n = ...
n
n
= a∗(ωkf)Θ−k, (36)
(cid:18)k(cid:19)
X
k=0
and similarly
n
n
Θna(f)Θ−n = (−1)k a(ωkf)Θ−k. (37)
(cid:18)k(cid:19)
X
k=0
Putting(36), (37)and (34) together,weobtain
n
n
[Θn,a∗(f)]Θ−n , [Θn,a(f)]Θ−n ≤ (ωk +1)f . (38)
(cid:18)k(cid:19) 2
(cid:13) (cid:13) (cid:13) (cid:13) Xk=1 (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:3)
Sinceτ 7→ H (τ)iscontinuousand[H (τ),Θ] = gQ⊗[a∗(f)+a(f),H ],
C SRC R
Lemma5and Theorem 4implythefollowingcorollary.
Corollary 6. Let L ∈ N and assume that ω−1/2f,ωL+2f ∈ L2(R3). Then τ 7→
0
H(g) (τ)Θ−1 is continuous and bounded, uniformly in τ ∈ R, and fulfills the
SRC
followingestimates,
ΘℓH Θ−(ℓ+j) ≤ kQk(M +M ), (39)
I −1/2 ℓ+j
(cid:13) (cid:13)
(cid:13)[Θℓ+j,H(g) (τ(cid:13))]Θ−(ℓ+j) ≤ gkQkM , (40)
SRC ℓ+j
(cid:13) (cid:13)
for allτ ∈ R, all ℓ(cid:13)∈ {0,1,...,L}, and j ∈(cid:13){1,2}. Moreover, τ 7→ H(g) (τ) isa
SRC
Kato-quasistablefamilyofself-adjointoperators,andtheunique,unitarysolution
of
∂ U(g) (t,s) = −iH(g) (t)U(g) (t,s), U(g) (s,s) = 1,
t SRC SRC SRC SRC
 (41)
∂ U(g) (t,s) = iU(g) (t,s)H(g) (s), U(g) (s,s) = 1,
s SRC SRC SRC SRC
obeys 
Θℓ+jU(g) (t,s)Θ−(ℓ+j) ≤ exp gkQkM (t−s) , (42)
SRC ℓ+j
(cid:13) (cid:13) (cid:2) (cid:3)
for all(t,s)(cid:13)∈ ∆, allℓ ∈ {0,1,...,L(cid:13)}, andj ∈ {1,2}. .