Decoherence of adiabatically steered quantum systems

0.34 MB·English

by  P. Solinas

Decoherence of adiabatically steered quantum systems
P. Solinas,1 M. Mo¨tto¨nen,1,2,3 J. Salmilehto,1 and J. P. Pekola2
1Department of Applied Physics/COMP, Aalto University, P.O. Box 14100, FI-00076 AALTO, Finland
2Low Temperature Laboratory, Aalto University, P.O. Box 13500, FI-00076 AALTO, Finland
3Australian Research Council Centre of Excellence for Quantum Computer Technology,
School of Electrical Engineering & Telecommunications,
University of New South Wales, Sydney, New South Wales 2052, Australia
We study the effect of Markovian environmental noise on the dynamics of a two-level quantum
system which is steered adiabatically by an external driving field. We express the master equa-
tion taking consistently into account all the contributions to the lowest non-vanishing order in the
coupling to the Markovian environment. We study the master equation numerically and analyti-
cally and we find that, in the adiabatic limit, a zero-temperature environment does not affect the
1
ground state evolution. As a physical application, we discuss extensively how the environment
1
affects Cooper pair pumping. The adiabatic ground state pumping appears to be robust against
0
2 environmental noise. In fact, the relaxation due to the environment is required to avoid the accu-
mulation of small errors from each pumping cycle. We show that neglecting the non-secular terms
n
inthemasterequationleadstounphysicalresults,suchaschargenon-conservation. Wediscussalso
a
a possible way to control the environmental noise in a realistic physical setup and its influence on
J
thepumpingprocess.
7

] I. INTRODUCTION usual treatment13 by means of Markovian master equa-
h
p tion non-trivial.14 Similar problems have been studied
- During the past three decades along with a greatly withdifferentapproaches15–17 inthe aimto estimate the
t
n increased interest in quantum mechanics and its appli- decoherenceeffectinducedbytheenvironment. Here,we
a use a particular master equation derived taking consis-
cations, we have witnessed a growing demand for accu-
u tently into account the drive, the environment and the
ratecontrolofquantumsystems. However,thesesystems
q
combinedeffect. Inparticular,weincludetheusuallyne-
[ are difficult to manipulate and very sensitive to external
noise. glected secular terms and find that, in our case, they be
2 handledwithextremecaresinceitcanleadtounphysical
A possible solution to the accurate control problem
v results. Our result that zero-temperature environment
may be the adiabatic manipulation. This approach not
7
does notaffectthe groundstate adiabaticdynamics sug-
4 only eliminates the need for fine tuning the time con-
geststhatgeometricquantumcomputationintheground
3 trol, but recently it has been proposed to solve com-
state18 couldberobustagainstdecoherence. Theanalyt-
2 plex computational problems1 and shown to be able to
7. produce any unitary transformation.2 Within the same icalderivationofthesolutionsofthemasterequationfor
different coupling and adiabatic evolutionis discussedin
0 framework,itisalsopossibletoperformnon-trivialperi-
detail. Similar results are obtained in the more realistic
0 odic control to manipulate the quantum system by geo-
case of finite temperature environment.
1 metricmeans. Thisapproachoriginatesfromthefamous
v: work of Berry3 but later it has been extended to more As a physical example, we study Cooper pair
i general cases4 and analyzed for its possible applications pumping19 and discuss in detail the effect of the en-
X in the so-called geometric quantum computing.5–8 The vironment in the pumping process. In this case, we
r main advantage is that the geometric dependence of the find that the physical observables are robust during the
a
applied unitary operator renders it robust against fast ground state pumping. However, neglecting the non-
fluctuationsinthecontrolsignal.9,10 Themaindrawback secular terms in the master equation leads to a decrease
of this proposalis that to exploit all the advantages,the in the predicted pumped charge and a difference in the
evolution time has to be long with respect to the time average charge pumped through the two junctions, i.e.,
scale of the system. However, the evolution time cannot charge non-conservation. These results are obtained an-
be increased indefinitely because the system undergoes alytically by direct calculation and confirmed by numer-
dephasing and relaxation leading potentially to serious ical analysis. We propose a way to implement in situ
problemsinmaintainingthequantumstate. Evenifsome the environmental noise by coupling a thermal resistor
efforts have been made to clarify the effect of the envi- to the system through an array of SQUIDs. The main
ronment and its possible elimination in the context of advantageofthisproposalisthat,bycontrollingtheflux
geometric quantum computation,11,12 a complete analy- throughtheSQUIDarray,wecanexperimentallychange
sis is still missing. the effective noise spectrum of the environment.
In this paper, we consider a two-level system initially Thispaperisorganizedasfollows. InSec. IIthemodel
in the ground state and study how it is influenced by is described. In Sec. III, we write the master equation,
the environment and an adiabatically changing exter- solveitinthequasi-stationarylimit,anddiscussdifferent
nal driving field. The time-dependent control makes the approximations. In Sec. IV, we apply the present model
2
to Cooper pair pumping in the presence of environment We define the local adiabatic parameter as α(t) =
and discuss different regimes, the effect of temperature, ~ w(t) /∆(t) where ∆(t) is the instantaneous min-
|| || | |
and the implication of the secular approximation. Fur- imum energy gap in the spectrum and w(t) =
|| ||
thermore,wedescribeawaytoengineertheenvironment Tr w†(t)w(t) is the tracenormofw(t). This instanta-
S
in situ,andcalculatethepumpedcharge. SectionVcon- neous adiabatic parameter gives an accurate estimate of
p
cludes the paper. thelocaldegreeofadiabaticityoftheevolution. Denoting
with T the time intervalfor the adiabaticevolutionand
p
setting ∆ =min[∆(t)], aroughestimateofαis given
min
II. MODEL by α¯ = ~/(∆ T ) which represents a global adiabatic
min p
parameter. The w(t) scales as 1/T and then, for adia-
p
We consider a quantum system subject to external batic evolution (α 1) it is usually neglected.21 How-
time-dependent control fields and interacting with the ever, as discussed in≪Ref.14, it can be crucial to include
environment. The total Hamiltonian is H(t) = H (t)+ its influence for the correct description of the dynamics
S
H +V, where H (t) denotes the time-dependent sys- of the system. In fact, the full Hamiltonian (1) suggests
E S
tem Hamiltonian with non-degenerate eigenvalues, H thatthe environmentleadstorelaxationto the instanta-
E
is the environment Hamiltonian and V is the system- neous eigenstates of H˜ (t)+~w(t). If we neglect w(t),
S
environment coupling. We assume that V = Y X, theenvironmentproducesrelaxationintheadiabaticba-
where X is the environment and Y is the system⊗part sis of H˜ (t) which usually differs from the correctone at
S
of the coupling operator. The simplest way to study the order O(α). As we will discuss in detail, this contri-
the dynamics of the system is to use the instantaneous bution can be important in determining the expectation
eigenbasis of H (t), i.e., the adiabatic basis composed of value of a physical observable.
S
states ψ (t) withenergiesE suchthatH (t) ψ (t) = III. MASTER EQUATION
n n S n
| i | i
E (t) ψ (t) . We denote by D(t) the transformation
n n
| i
from a given fixed basis to the adiabatic one. To prop-
erly take into account the time-dependence of the basis,
we study the dynamics of ρ˜ = D†(t)ρ (t)D(t) in the
tot tot
fixed basis. Here ρ (t) is the usual density operator of Letusstudythedynamicsofagenerictwo-levelsystem
tot
the total system in the Schr¨odinger picture. The evolu- steeredby atime-dependent Hamiltonianandsubjectto
tion of ρ˜ is governed by the effective Hamiltonian environmental noise. If the evolution is adiabatic and
tot
the system interacts weakly with the environment, we
H˜(1)(t)=H˜S(t)+~w(t)+V˜(t)+HE, (1) can treat ~w(t) + V˜(t) as a perturbation and derive a
masterequationforthe densitymatrixinthe interaction
where H˜S(t) = D†(t)HS(t)D(t), V˜(t) = D†(t)VD(t) = picture.14 Up to the first order in α and the second in
Y˜(t) X, and w = iD†(t)D˙(t). the system-environment coupling, we obtain
⊗ −
dρ˜ (t) 1 t
I =i[ρ˜ (t),w (t)] Tr dt′ [ρ˜ (t) ρ ,V˜ (t′)],V˜ (t) +
dt I I − ~2 E I ⊗ E I I
(cid:26)Z0 (cid:27)
i t t′ (cid:2) (cid:3)
+ Tr dt′ dt′′ ρ˜ (t) ρ ,[w (t′),V˜ (t′′)] ,V˜ (t) , (2)
~2 E(Z0 Z0 h (cid:2) I ⊗ E I I (cid:3) I i)
where ρ˜ (t) is the density matrix of the system in tributionontherightisthedrivingtermwithoutsystem-
I
the interaction picture, Tr indicates trace over the environment interaction, the second one leads to the
E
environmental degrees of freedom and ρ is the sta- standard dissipative contribution of the Bloch-Redfield
E
tionary density operator of the environment. The in- theory. The third contribution is a cross-term of the
teraction picture operators are defined as O˜ (t) = driveanddissipation. This lasttermisusuallyneglected
I
eiHEt/~US†(t,0)O˜(t)US(t,0)e−iHEt/~, where US(t,0) = as discussed above.22
e−i/~R0tH˜S(τ)dτ is the time-evolution operator, with Denoting the ground and excited state of HS as g
T T | i
denoting time-ordering. In this frame, we can address and e , respectively, the master equation arising from
| i
andinterpretthe contributionsofEq. (2). Thefirstcon- Eq. (2) in the Schr¨odinger picture assumes the form
3
ρ˙ = 2 m(w∗ ρ )+S(ω ) m 2 [S( ω )+S(ω )] m 2ρ +2[ m(m ) m(ρ )+ e(m ) e(ρ )]S(0)m
gg − ℑ ge ge 0 | 2| − − 0 0 | 2| gg ℑ 2 ℑ ge ℜ 2 ℜ ge 1
2S(0) S( ω ) S(ω )
0 0
2 − − − [ m(m ) m(w )+ e(m ) e(w )][ m(m ) m(ρ )+ e(m ) e(ρ )]
2 ge 2 ge 2 ge 2 ge
− ω {ℑ ℑ ℜ ℜ ℑ ℑ ℜ ℜ }
0
2S(0) S( ω ) S(ω )
0 0
+2 − − − m m(m ) m(w )+ e(m ) e(w ) ρ
1 2 ge 2 ge gg
ω {ℑ ℑ ℜ ℜ }
0
S(0) S(ω )
0
2 − m m(m ) m(w )+ e(m ) e(w ) , (3)
1 2 ge 2 ge
− ω {ℑ ℑ ℜ ℜ }
0
and
ρ˙ = iw (2ρ 1)+i(w w )ρ +iω ρ S(ω )m m +[S( ω )+S(ω )]m m ρ 2S(0)m2ρ
ge ge gg − ee− gg ge 0 ge− 0 1 2 − 0 0 1 2 gg − 1 ge
i[S( ω )+S(ω )]m [ e(m ) m(ρ ) m(m ) e(ρ )]
0 0 2 2 ge 2 ge
− − ℜ ℑ −ℑ ℜ
2S(0) S( ω ) S(ω )
2 − − 0 − 0 m2w ρ
− ω 1 ge gg
0
S(0) S(ω ) S( ω ) S(ω )
+2 − 0 m2w im − 0 − 0 m(m ) e(w ) e(m ) m(w )
ω 1 ge− 2 ω {ℑ 2 ℜ ge −ℜ 2 ℑ ge }
0 0
2S(0) S( ω ) S(ω )
0 0
2 − − − m im [ m(w ) e(ρ ) e(w ) m(ρ ))]
1 2 ge ge ge ge
− ω { ℑ ℜ −ℜ ℑ
0
[ m(m ) m(w )+ e(m )s e(w )]ρ , (4)
2 ge 2 ge ge
−ℑ ℑ ℜ ℜ }
where ~ω is the (instantaneous)energy gapbetween g thepositivityofthedensitymatrix. Forthisreason,par-
0
and e , m =Y (t)/~= Y (t)/~,m =Y (t)/~,a|ndi ticular attention must be paid to the parametersused in
1 gg ee 2 ge
| i −
S( ω ) and S(0) are obtained from the power spectrum the numerical simulations since they can lead to inaccu-
0
oft±henoiseS(ω)= ∞ X (τ)X (0) eiωτdτ.23Here,the rateresultsinthenon-adiabaticcaseandstrongcoupling
−∞h I I i
matrix element of a generic operator O is denoted by regime.
O = k Ol (withRk,l =g,e) except for w = i k l˙ .
kl kl
h | | i − h | i
Noticethatthepresentequationsgeneralizetheonespre-
sented in Ref.14 because they are obtained for a generic
system-environment coupling operator.
A. Secular Approximation
In the derivation of equations (3) and (4), we have
implicitly assumed that our system is in the Markovian
InEqs. (3)and(4), we haveincluded allthe contribu-
regime. This implies that the bath correlation time τ is
short compared to the relaxation time of the system. In tions up to the second order in α and V contrary to the
addition, since the evolution is adiabatic, ω , m , m , usually adopted secular approximation.13 In the interac-
0 1 2
and w change slowly in time and we can approximate tion picture, the non-secularterms oscillate rapidly with
w(t+τ) w(t). respect to the relaxation-dephasing dynamics generated
≈ by the environmentandthus their contributionaverages
In writing the above equation we have neglected the
correctionsduetotheLambshift. However,weexpectit to zero. Even though this approximation does not lead
toproblemsinmanycases,weshowthatitisinadequate
to have smallinfluence onthe dynamics ofthe systemin
the adiabatic limit. A detailed analysis is under way.24 in describing adiabatic evolution.
Notice that Eqs. (3) and (4) are, strictly speaking, To compare the results with and without these terms,
valid only for adiabatic evolution and weak system- weexplicitlywritethemasterequationscorrespondingto
environment coupling. In addition, they are not in the Eqs. (3) and (4) but obtained with the secular approxi-
standardLindbladformandtheydonotaprioriconserve mation as
4
ρ˙sec = 2 m(w∗ ρ )+S(ω ) m 2 [S( ω )+S(ω )] m 2ρ
gg − ℑ ge ge 0 | 2| − − 0 0 | 2| gg
2S(0) S( ω ) S(ω )
0 0
+ 2 − − − m(m ) m(w )+ e(m ) e(w ) m ρ
2 ge 2 ge 1 gg
ω {ℑ ℑ ℜ ℜ }
0
S(0) S(ω )
0
2m − m(m ) m(w )+ e(m ) e(w ) , (5)
1 2 ge 2 ge
− ω {ℑ ℑ ℜ ℜ }
0
and
1 1
ρ˙sec = iw (2ρ 1)+i(w w )ρ +iω ρ [ S( ω ) m 2+ S(ω ) m 2+2S(0)m2]ρ
ge ge gg− ee− gg ge 0 ge− 2 − 0 | 2| 2 0 | 2| 1 ge
2S(0) S( ω ) S(ω )
+ − − 0 − 0 (2m w +w m∗)m ρ . (6)
ω 2 eg ge 2 1 ge
0
Noticethat,inthetime-independentbasis(i.e. w 0), In the following, we consider first the finite tempera-
kl
≡
these reduce to the usually adopted Bloch equations for ture case, in which we assume that the excitation rates
a non-driven system are exponentially small with respect to the relaxation
ones;forexample,Γ =Γ exp[ ~ω /(k T)], whereT
ge eg 0 B
ρ˙Bgg = Γ↓ρee−Γ↑ρgg, is the temperature of the bath. −To the lowest order in
ρ˙B = iω ρ Γ ρ , (7) α and in the quasi-stationary limit, the solution for ρgg
ge 0 ge− ge ge has a simple expression
where Γ = S(ω ) m 2, Γ = S( ω ) m 2,
and Γge =↓ 21S(−ω0)|0m|2|22+| 21S(ω↑0)|m2|2+−2S0(0|)m221| . ρgg = Γ Γ+egΓ +O(α2)≈1−e−k~BωT0 +O(α2). (10)
These equations have immediate interpretation in terms eg ge
(cid:2) (cid:3)
of relaxation and dephasing processes.
The system remains in the ground state in the linear or-
derinα,apartfromleakageduetothefinitetemperature
environment.
B. Adiabatic quasi-stationary solutions for the
ground state and the role of secular approximation Within the same approximation and using the above
result, the equation for the off-diagonal term reads
In absence of system-environment interaction, i.e.,
ρ˙ = (iω 2S(0)m2)∆
S(ω) = 0, Eqs. (3) and (4) become the standard von ge 0− 1 ge,1
Neumann equations for the density matrix + i m(m )m S(ω ) e(∆ )
2 2 0 ge,2
ℑ ℜ
i e(m )m S(ω ) m(∆ )=0,
ρ˙ = 2 m(w∗ ρ ), − ℜ 2 2 0 ℑ ge,2
gg − ℑ ge ge (11)
ρ˙ = iw (2ρ 1)+i(w w )ρ +iω ρ .(8)
ge ge gg ee gg ge 0 ge
− −
where we have defined
Since we are interested in the evolution in the quasi-
fsotarttihoneasroylultimiointofofrρ˙aggdi=ab0ataicndevρo˙gleut=ion0.(Wαe≪h1a)v,ewe look 1−2e−k~Bω0T wge
∆ = +ρ ,
ge,1 (cid:16) ω (cid:17) ge
ρ = 1+O(α2), 0
gg −~ω0
ρge = −wωg0e +O(α2). (9) ∆ge,2 = 1+e−k~BωT0 ρge+ (cid:16)1−e ωk0BT(cid:17)wge.(12)
(cid:16) (cid:17)
Ingeneral,Eqs. (3)and(4)canbeintegratedonlynu-
At low but finite temperatures, the solution of Eq. (11)
merically. However,wecanobtainanalyticalinformation
is
ifweexploittheadiabaticlimit. Asabove,thecorrection
totheground-statepopulationρgg isoforderα2 andthe wge −~ω0
off-diagonal element ρge scales as α. ρge =− ω0 (1−2e kBT) . (13)
5
Notice that, in the zero-temperature limit, solutions in the ground and the excited state are equally populated,
Eqs. (10) and (13) coincide with the the environment- i.e., ρ = ρ = 1/2. With this assumption, the differ-
gg ee
free solutions (9): the ground-state evolution is robust ential equation for ρ in (8) can be solvedto obtain the
ge
andnotinfluencedbythezero-temperatureenvironment. quasi-stationary solution
This can be explained in a simple way.14 The effective
Hamiltonian in the absence of system-environment cou- ρgg = 1/2,
plingisH˜S+~w. Intheadiabaticlimit, thesecondterm ρge = 0 (15)
canbetreatedasaperturbationandthecorrectiontothe
eigenstatesofH canbecalculated. Thisbasisisusually apartfromcorrectionsofthesecondorderinthesystem-
S
called superadiabatic.22 Up to the linear order in α, the environment coupling. The system undergoes sequential
superadiabatic ground state is g′ = (g w∗ /ω e ). transitionsfromthe groundto the excitedstate andvice
| i | i− ge 0| i
In the adiabatic evolution, if the system starts in the versa forming a fully mixed state.
groundstate g′(0) ,wecanassumethatitremainsinthe Thus, in the two limits in which relaxation dominates
eigenstate g′(|t) anidthecorrespondingdensitymatrixis non-adiabatic transitions and non-adiabatic slight tran-
ρ′ = g′(t)| g′(t)i. Thus, the density matrix elements in sitions dominate relaxation,it is possible to find an ana-
the H| baishis, ρ| = g ρ′ g and ρ = g ρ′ e satisfy lyticalquasi-stationarysolutionwhichcorrectlydescribes
S gg ge
h | | i h | | i
Eq. (9). thesystembehavior[Eqs.(9)and(15),respectively]. No
This result helps to understand the robustness of such solution is available in the intermediate regime in
ground-stateevolutionsubjecttozero-temperaturenoise. which relaxation and non-adiabatic transitions occur on
In this regime, the environment cannot produce transi- the same timescale.
tions towards the excited state e′(t) but induces only
relaxation to the ground state g|′(t) .i Since the system
| i
starts from the ground state and evolves adiabatically, IV. APPLICATION TO COOPER PAIR
it remains always in the ground state of H˜ +~w and PUMPING
S
the environment has no influence on its dynamics. In
this framework,it is clearthat neglecting the w(t) terms In the previous section, we showed that ground-state
leads to relaxation with respect to the basis of H˜ and evolutioncanberobustagainstenvironmentalnoise. The
S
hence to different dynamics. information about the dynamics of the system is con-
The above result demonstrates the importance of the tainedinthe off-diagonaltermofthe density matrix,see
choice of the proper master equation. To emphasize this Eq. (13). Thus to test our theoretical model, it is nat-
pointfurther,weuseasimilarapproachtosolvethemas- ural to focus on a physical observable which depends on
terequationarisingfromthesecularapproximation[Eqs. ρ . An interesting candidate for the observable is the
ge
(5) and (6)]. In particular,in the zero-temperaturelimit pumped charge in a superconducting circuit. Recently,
and up to the order of the product of α and the square it has been subject to intense study both theoretically25
of the system-environment coupling, we obtain and experimentally26 because of its connection to ge-
ometric phases,25,27,28 and its potential application in
ρsec = 1, quantum metrology.19,20,30
gg
2iw In particular, we consider the Cooper pair sluice19
ρsgeec = −2iω geΓ, (14) shown in Fig. 1. It consists of a single superconducting
0− island, coupled to superconducting leads via two super-
where Γ = S(ω ) m 2 +4S(0)m2 and the corrections conductingquantuminterferencedevices(SQUIDs). The
0 | 2| 1 SQUIDs operate as Josephson junctions whose critical
are of order of the product of the adiabatic parameters
currents can be tuned by magnetic fluxes. The electro-
andthesquareofthesystem-environmentcoupling. The
static potential on the island can be controlled by vary-
expression of the diagonal term is not influenced by the
ing a gate voltage, V , and there is a constant supercon-
secular approximation,but the off-diagonalone depends g
ducting phase difference, ϕ = ϕ +ϕ between the two
onthecouplingbetweenthesystemandtheenvironment. R L
leads. The operator n = i∂ (k = L,R) represents
This is in contrast with both the physical picture dis- k − ϕk
the Cooper pair number operator of the k-th SQUID. In
cussed above and analytical results in Eq. (13). This
the absence of noise, the Hamiltonian of the sluice can
difference leads to different expectation values for physi-
be expressed as14
cal observables.
We can obtain analytical information also in the op- ϕ ϕ
posite regime in which the evolution is fast enough that HS =EC(n ng)2 JLcos( θ) JRcos( +θ). (16)
− − 2− − 2
non-adiabatic transitions determine the dynamics of the
system. In this limit, neglecting the relaxation terms Here θ =(ϕ ϕ )/2 andn= i∂ are the operators
R L θ
− −
in Eqs. (3) and (4), we obtain again Eq. (8). The sys- for the superconducting phase and the number opera-
tem is initially in the ground state with ρ (0) = 1 and tor of excess Cooper pairs on the island. The Josephson
gg
ρ (0) = 0. We expect that, due to sequential non- couplings to the left and right lead are denoted by J
ge L
adiabatic excitations, in the quasi-stationary limit both and J , respectively, n = C V /(2e) is the normalized
R g g g
6
(a)
(b)
J M
L
J m
L
J M
R
J m
R
n M
g
n m
g
0 t(cid:144)Tp 1
FIG.1: (Coloronline)(a)CircuitschemeoftheCooperpairsluice(physicalrealization ofanadiabatictwolevelsystem). The
time-dependent control parameters are JR, JL, and ng. The phase difference across the device is denoted by ϕ. The circuit
in the dashed box is used to engineer the environmental noise characterized and modeled by a resistor R, a voltage δVE and
coupledtothesystemwithaneffectivecapacitanceCE. (b)Thetime-dependenceoftheparameters{JL,JR,ng}inapumping
cycle.
gate charge,and E =2e2/C is the charging energy of Its average value reads
C Σ
the sluice; C is the gate capacitance and C the total
g Σ
capacitanceoftheisland. Forfurtherconvenience,wede- I =(ρ I +ρ I )+2 e(ρ I ), (18)
k gg k,gg ee k,ee ge k,eg
ℜ
finethedeviationofthegatechargefromthedegeneracy
point as δng =ng−1/2. where Ik,nm = hn|Ik|mi with m,n = g,e. The last
The average-value given by the current through the term is the geometric contribution corresponding to the
k-th SQUID is Ik = 2eTr(ρ˙totnk) where the trace is pumped charge
−
over the degrees of freedom of both the system and the
environment, and ρtot is the total density matrix of the Tp
QG = 2 e(ρ I )dt. (19)
system and the environment. Using the von Neumann k ℜ ge k,eg
equation for ρ , we obtain Z0
tot
IfE max J ,J andn 1/2,onlythetwolow-
2ie C L R g
≫ { } ≈
Ik = ~ iTr(ρtot[nk,HS(t)])+Tr(ρtot[nk,HE]) est charge states are important for the dynamics and we
can adopt the two-state approximation. Let 0 and 1
+Tr(ρtot[nk,V]). (17) denote the states with no and one excess Coop|eir pair|oni
the island, respectively. Thus the coupling between the
Writingthe complete traceintermsofpartialtracesand
sluice and charge noise has the form V =egσ δV (t),
tracing out only the environmental degrees of freedom, z⊗ E
where σ = 0 0 1 1 a part terms proportional to
weobtainTrS(ρ[nk,HS(t)])whichistheusualdefinition z | ih |−| ih |
identity.
of the current for a closed system.27 The second term
gives no contribution since n and H commute. The
k E
thirdtermisthecurrentdirectlyinducedbytheenviron-
ment. Notice that, the last contribution is present only A. Gate charge noise
if[n ,V]=0. Inallthe othercases,wecanusethestan-
k
dard defin6 ition of current operator: Ik = 2~ei[nk,HS(t)]. With the above notations and in this approximation,
We study the case in which the noise is due to gate we canexpress all the relevant quantities in terms of the
voltage fluctuations coupled capacitively to the sluice control parameters J , J and n as
L R g
island.31 The charging part of the Hamiltonian in Eq.
(16) has the form EC[n Cg/(2e)Vg]2. If an addi- 1
tional noisy gate potentia−l δVE is coupled to the sys- E12 = 2 JL2 +JR2 +2JLJRcosϕ,
tem through a capacitance CE, we can use the above q J J ϕ
R L
expressiontowritetheHamiltoniandescribingtheinter- γ = arctan − tan ,
J +J 2
action between the system and the environment. At the (cid:18) R L (cid:19)
lowest order in δVE the Hamiltonian can be written as η = δng ,
V = 2egn δVE(t), where g = CE/CΣ is the coupling δn2+(E12)2
const−ant bet⊗ween the system and environment. Notice g EC
that since [nk,V] = 0, the current operator of the kth ω = q2E12 , (20)
SQUID is determined only by the first term in Eq. (17). 0 ~ 1 η2
−
p
7
(a) (b)
1.05
1.00
f
1.00
e e 0.95 0.4
2 0.95 2
(cid:144)QG g (cid:144)QG 0.90 HLΑt 0.2
0.90
0
0.85 0.85 0 0.5 1
t(cid:144)T
p
0.80
0.80
-3 -2 -1 0 1 2 3 0.005 0.025 0.05 0.075 0.1
j g
FIG. 2: (Color online) Pumped charge per cycle in a Cooper pair sluice for zero-temperature environment (a) as a function
of phase ϕ with coupling constant g = CE/CΣ = 0.01,0.0125,0.015,0.1 from bottom to top for f = 1/Tp = 75 MHz, and
(b) as function of coupling g with ϕ=π/2 and at frequencies f = 10,25,50,65,75 MHz from bottom to top in the region in
which the arrow is plotted. In the numerical simulations, the parameters are JiM/EC = 0.1, Jim/JiM = 0.03 (with i = L,R),
δnMg = −δnmg = 0.3, EC/kB = 1 K [EC/(2π~) = 21 GHz], R = 300 kΩ, environment temperature T = 0, S(ω0) = 2~ω0R,
S(−ω0) = 0, and S(0) = 2kBT0R with T0 = 0.1 K. In inset we show the time evolution of the adiabatic parameter α(t) for
f =75 MHz.
and the instantaneous eigenstates assume the form In our simulations the spectral density is taken to be
ohmic suchthat S(ω)=2~ωR/[1 e−~ω/(kBT)] where R
1 −
g = ( 1 η 0 +e−iγ 1+η 1 ), istheeffectiveresistanceofthe environmentalnoise. For
| i √2 − | i | i finite temperatures, we have S( ω) = e−~ω/(kBT)S(ω)
1 p p and S(0) = 2k T R, where T i−s the effective tempera-
e = ( 1+η 0 e−iγ 1 η 1 ). (21) B 0 0
| i √2 | i− − | i ture of the dephasing process. We note that the master
p p equations presentedin this paper are also validfor other
Further, the matrix elements of Y (V = Y X) read types of environment spectra.
⊗
~m = gη and ~m =g 1 η2.
1 2
− −
The w matrix elements we need are
p
B. Numerical Results
1
w = (1+η)γ˙
gg
−2
The pumping cycle taken into consideration is shown
1
w = (1 η)γ˙ inFig.1(b)andthecorrespondingevolutionofthequan-
ee
−2 −
tum system is obtained by numerical integration of Eqs.
1 η˙
w = ( 1 η2γ˙ i ) (22) (3) and (4). The physicalobservable here is the pumped
ge
2 − − 1 η2 charge through the junction in the stationary regime
p −
whichis reachedafter severalsequentialpumping cycles,
Using I = 2e∂ H /~,27 the cuprrent operators are
k ϕk S i.e., after the initial transient is over. In fact, during the
I = (2e/~)J sin(ϕ θ) and I = (2e/~)J sin(ϕ +
L L 2 − R R 2 first cycles the pumped charge oscillates due to simul-
θ). We can write the current operators restricted to the
taneous effects of non-adiabatic excitation and environ-
0 , 1 basisusingtheformulaseiθ = 1 0 ande−iθ =
mental relaxation and it takes several cycles to stabilize
{| i | i} | ih |
0 1. Inserting them in Eq. (19), the charge pumped
to the stationary values which are measured in the ex-
| ih |
through each junction reads
periments.
InFig.2(a),thepumpedchargeisshownasafunction
2e Tp ϕ
QG = dtJ η e(ρ )sin γ+ ofthephaseϕacrossthedeviceϕandfordifferentvalues
L ~ Z0 Lϕ(cid:16) ℜ ge (cid:16) 2(cid:17) of the coupling strength g while the pumping frequency
is kept constant. The ideal pumped charge in the ab-
cos γ+ m(ρ )
− 2 ℑ ge sence of noise19 is also shown and it coincides with the
2e T(cid:16)p (cid:17) ϕ(cid:17) numerical result in the strong coupling limit. For weak
QG = dtJ cos γ m(ρ )
R ~ R − 2 ℑ ge coupling to the environment, the pumped charge differs
Z0 (cid:16) (cid:16) ϕ (cid:17) substantially from the ideal one but an increment of the
η e(ρge)sin γ . (23) system-environmentcouplingresumestheidealpumping.
− ℜ − 2
(cid:16) (cid:17)(cid:17) In Fig. 2(b), we present the numerically simulated
We define the averagepumped chargeas QG (QG+ pumped charge as a function of g, for different pumping
QG)/2andchargeconservationimpliesthatQG≡=QGL = frequencies. For an adiabatic loop (e.g. f = 1/T = 10
R L p
QG. MHz),thepumpedchargeisclosetotheidealvalue. This
R
8
(a) (b) 1.00
1.00
0.95
0.95
n n
o 0.90 o
ulati 0.85 ulati 0.90 g
op 0.80 f op
P P
0.85
0.75
0.70
0.80
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
t(cid:144)T t(cid:144)T
p p
FIG.3: (Coloronline)PopulationofthegroundstateofH˜S+~winthequasi-stationaryevolution. (a)Evolutionwithg=0.01
and f = 10,75,100 MHz from top to bottom. (b) Evolution in presence of environmental noise for a non-adiabatic 75 MHz
cycle with g=0.01,0.0125,0.015 from bottom totop. The otherparameters are thesame as in Figure 2.
confirmsthefirstanalyticalresults: intheadiabaticlimit does not guarantee a priori the positivity of the density
and T = 0, the environmental noise does not influence matrixandthatwehavetruncatedatthefirstorderinα
the evolution. For faster pumping frequencies, the be- and g in the derivation of the master equation. Strictly
havior is more complex because it is determined by the speaking the g and α used are always small compared
competing effects of non-adiabatic transitions and envi- to 1 [see Fig. 2(b)] and thus, at least formally, they can
ronmental relaxation. When the non-adiabatic transi- be used as expansion parameters. Unfortunately, such
tions dominate over the relaxation process the predicted analysis gives only an estimate of the corrections we are
pumped charge is smaller than the ideal one. This is neglecting and proper analysis should include the higher
due to the fact that the excited state e′ of H˜ +~w order terms.29
S
| i
is populated during the evolution and it pumps in the
opposite direction with respect to the ground state g′ . However,there are indications thatthe features in the
Increasingthe coupling we enter the regimein which|thie αandg regimespresentedarerealistic. The convergence
relaxationprocess dominates. Here, the system is forced to the ideal solution predicted by the analytical analysis
to remain in the ground state g′ and we restore the is present in the simulations for all the range of param-
ideal pumping. Thus, the numer|icail simulations confirm eters used. It seems unlikely that such common behav-
the analytical results obtained in Sec. IIIB for different ior, which is realistic for small α and g, is a numerical
value of α/g ratio. artifact. The violation of positivity, which has a role
To check this interpretation of the numerical results, in the intermediate regime, becomes less important for
we calculate the population of g′ during the pump- the maximum value of g presented. In fact, the analy-
ingcyclesfordifferentenvironmen|tailcouplingstrengths. sisperformedconfirmsthatintheseextremeregimesthe
This is obtained by solving Eqs. (3) and (4) numeri- positivity tends to be restored.
cally and projecting the obtained density matrix into
the g′ , e′ basis. Figure 3(a) shows how increasing
the f{r|eqiue|nciy}leads to a leakage from the ground g′ to
the excited e′ state. The figure shows that incr|easiing
| i
the coupling strength from the region g α 1 to
≪ ≪
α g 1, the system begin to remain in the ground
sta≪te g≪′ , thus restoring the evolution expected in the C. Finite Temperature Pumping
| i
adiabatic limit without noise. This is a generic feature
and we have chosen 75 MHz pumping frequency to illus-
trate the phenomenon. InFig.4,wecompareCooperpairpumpingwithzero-
Even if the convergence to the ideal evolution has and finite-temperature environments. We observe that
strong physical motivation, we stress that the details of the features of the zero-temperature simulation are still
the numerical simulations in the extreme regime must presentalthough,atfinite-temperature,weobtainingen-
be taken with care. In fact, in the intermediate regime eral a smaller pumped charge with the same parame-
boththeanalyticalandnumericalapproachesfailtopre- ter values. This is expected since the presence of finite-
dict the correct behavior of the pumped charge. In this temperature environment leads to an additional leakage
regime, the numerical simulations in Fig. 2(b) are in- from the ground state to the excited state, leading to a
accurate and present an excess of pumped charge with reduced pumped charge since the adiabatically pumped
respect to that in the adiabatic limit. This unphysical chargein the excitedstate is equalin magnitude but op-
prediction is due to the fact that the master equation posite in sign to that in the ground state.27
9
(a) (b)
1.05
1.00
f
1.00
0.95
e e
2 0.95 2
(cid:144) (cid:144)
QG g QG 0.90
0.90
0.85
0.85
0.80
0.80
-3 -2 -1 0 1 2 3 0.005 0.025 0.05 0.075 0.1
j g
FIG. 4: (Color online) Pumped charge per cycle for finite-temperature environments (dashed curves) in comparison to the
zero-temperature environment case (solid lines): (a) as a function of phase ϕ for f =75 MHz and g = 0.01,0.0125,0.015,0.1
from bottom to top and (b) as a function of the coupling strength g with ϕ = π/2 and frequency f = 10,50,75 MHz from
bottom to top. The other parameters are as in Fig. 2 except for T =0.03 K for thefinite temperaturesimulations.
D. Pumped charge with secular approximation and terms, we obtain a result which is essentially immune to
charge conservation g. Importantly, if the non-secular terms are included in
the master equation, the two charges through the left
Alltheresultspresentedabovedependonwhetherone and right junctions are equal but if we apply the secular
includes in the master equation the non-secular terms approximation,weobtainasignificantdifferenceinthese
andthecombinedeffectoftheenvironmentandthedriv- charges. The difference leads to chargenon-conservation
ing or not . From the analyticalcalculation presented in since nochargecanbe accumulatedonthe islandfor the
Sec. IIIA, we know that, in the quasi-stationary limit, quasi-stationary solution.
the off-diagonal elements of density matrix within the We can also estimate analytically this charge non-
secular approximation [Eq. (14)] have a different form conservation. Let us define the charge asymmetry be-
compared with the full expression [Eq. (13)]. Thus we tween the junctions as ∆Q = QG,sec QG,sec, where
L − R
expect this to lead to different values of physical observ- QG,sec is the charge through junction k described
k
ables. To compare the two methods, the pumped charge by Eq. (14). Using Eqs. (20), (23), and J =
R
is again a privileged candidate since it depends directly J sin γ+ ϕ /sin γ ϕ , we obtain
− L 2 − 2
on the off-diagonal matrix element ρ .
ge
In the quasi-stationary limit, within the secular ap- (cid:0) (cid:1)2e (cid:0)Tp (cid:1)~η˙
proximation, at zero temperature, and for weak system- ∆Q = ~ dt 2 −2E12ℑm(δρ) . (25)
environment coupling, we obtain from Eq. (14) Z0 (cid:18) (cid:19)
The first term in the integralgives no contribution since
w iΓ Γ2
ρsec ge 1 =ρ δρ, (24) η(0) = η(T ). This is itself an important result since it
ge ≈− ω − 2ω − 4ω2 ge− p
0 (cid:18) 0 0(cid:19) means that, if we keep the non-secular terms (i.e. δρ =
0), we obtain immediately that the charge is conserved.
where δρ denotes the correction with respect to the full
solution. From this expression it results that ρsec is With the help of Eq. (24), the second term can be
ge
explicitly written as
dampedbecause ofthe interactionwith the environment
and this effect should reduce the pumped charge. To
e(w )Γ m(w )Γ2
demonstrate this effect, we have chosen an asymmetric ge ge
m(δρ) = ℜ + ℑ . (26)
pumpingloopinthefluxandgatevoltageparametersand ℑ 2ω2 4ω3
(cid:18) 0 0 (cid:19)
performed a numerical integration of Eqs. (3), (4), (5),
and(6). Theresultswithandwithoutthesecularapprox- UsingEqs.(20),(22),andthedefinitionofΓandS(ω)
imationarepresentedinFig. 5. Thepumpedchargepre- at zero-temperature given in Sec. IIIB and IVA, re-
dictedwiththesecularapproximationdecreaseswhenthe spectively, we can write explicitly the chargeasymmetry
coupling increases whereas if we include the non-secular as a function of time-dependent parameters
10
∆Q = 2eg2~ Tpdt[ γ˙ 1R 1 η2 2+ 1−η2 3/2S(0)η2
Z0 − 2 − (cid:0) 2(cid:1)E12 !
(cid:0) (cid:1)
+η˙g2~ 1R2 1 η2 2 η2−1 S2(0)η4 + R 1−η2 3/2S(0)η2 ]. (27)
2 − − (cid:0) 2E(cid:1)122 (cid:0) E(cid:1)12 !
(cid:0) (cid:1)
ing contributions
QG
0.8 L
[JmJM +JmJM] (δnM)3 δnm 3 g2~2S
A = L R R L g −| g | 0
0.7 1 16(E δnmδnM)3
e (cid:2)C g g (cid:3)
2
(cid:144) Jm(JM)3+Jm(JM)3 (δnM)4 (δnm)4 g2R~2
QG 0.6 secular A2 = (cid:2) R L L96R(ECδ(cid:3)n(cid:2)mg δngMg )4− g (cid:3)
0.5 (JM)2 (JM)2 (δnm)4+(δnM)4 g4~3S2
QGR A4 = L − R64(E δngmδnM)4 g 0
0.4 (cid:2) (cid:3)(cid:2)C g g (cid:3)
0.025 0.05 0.075 0.1 (JM)4 (JM)4 (δnm)5+ δnM 5 g4~3S2
A = L − R g | g | 0, (29)
g 5 128(E δnmδnM)5
(cid:2) (cid:3)(cid:2)C g g (cid:3)
FIG. 5: (Color online) Pumped charge obtained using differ- where ∆Q = i=1,5Ai with A3 = 0. The first two
ent master equations presented for f = 10 MHz cycle. The contributions are present only if the loop is asymmetric
straightlineisthesolutionofthefullmasterequation(3)and in the δng parPameter (i.e. δnMg 6= |δnmg |) whereas A4
(4), the solid lines are the solutions of the master equation and A yield non-vanishing contributions if the loop is
5
with secular approximation (5) and (6). In the latter case, asymmetric in the flux parameters (i.e. JM = JM).32
thecentral solid line is theaverage pumped charge while the If the loop is asymmetric in both parameteLrs a6s thRe one
other two lines are the pumped charges through the left and used in the simulations of Fig. 5, all the A terms in
i
right junctions. Here, the parameter cycle is strongly asym-
Eq. (29) contribute to ∆Q.
metric with JLM/EC = 0.1, JLm/JLM = 0.006, JRM/EC = 0.2,
Jm/JM = 0.04, δnM = 0.4 δnm = −0.03, T = 0, and the
R R g g
other parameters are as in Fig. 2.
E. Environment engineering
Theexperimentalset-updescribedinthispaperhasal-
In this expression, we can address five contributions.
readybeen usedfor the measurementofthe Berryphase
Sincethethirdtermdependsonlyonηanditsderivative, in adiabatic pumping of Cooper pairs.26,27 To study the
ityieldsnocontributionwhenintegratedalongthecycle.
effects of the environment in the pumping, it is bene-
Theremainingtermscanbecalculatedifwefixthephase
ficial to control the coupling between the system and
to ϕ=π/2 and in the regime in which Jm JM E
and δnM, δnm JM/E . Using Eqs.i (2≪0) aind≪(22C), its environment. This can be achieved by adding to the
g | g | ≫ i C already present intrinsic environment an experimentally
wecanwriteEq.(27)intermsoftheexperimentallycon-
controlledartificialnoisesource. Similarapproacheshave
trolled parameters J , J , and n , and take advantage
L R g been developed in different physical systems for both
of the control cycle in Figure 1(b). To have an idea of
testing the robustness of the studied quantum system
how the final terms look like we write the first term in
and the cross-over between the quantum and classical
Eq. (27) as
behavior.33,34 Here, we present a way to engineer in situ
the environment noise in the superconducting circuits
Tp J˙RJL J˙LJR η2(1 η2)32g2RS0 discussed above.
A = dt − − . The environment is controlled by the circuit in Fig.1
1 Z0 (cid:16) 2(JL2(cid:17)+JR2)32  which is composed of a thermal resistor R with noise
 (28) voltage spectrum δVE. We place an array of m SQUIDs
It is convenientto divide the integrationin six parts [see betweentheresistorandtheislandofthesluiceasshown
Fig.1(b)]: forexample,inthefirstpathonlyJ depends in Fig. 6. The array can be represented as a series of
L
ontime. Theremainingη functioncanbeapproximated, RLC parallel circuits with resistance RS, inductance L,
ssiwnictec,hifnogr Jbiemtw≪eenJiM1≪[seEeCE,q.it(t2e0n)d].s to a box function aZndcap=acitanceRCSZSC.STZhLeimped=anceofaisLin(φg)lωeRSSQUIDis,
± RLC RSZCS+ZLZCS+RSZL iL(φ)ω+RS(1−L(φ)ω2CS)
With this further approximation,we obtain four lead- where L(φ) = L /cos(πφ/Φ ), Z and Z are the
0 0 L CS