Table Of ContentComplete All-Optical Quantum Control of Electron Spins in InAs/GaAs Quantum
Dot Molecule
Guy Z. Cohen
Department of Physics, University of California, San Diego, La Jolla, California 92093-0319
(Dated: January 9, 2015)
ThespinstatesofelectronsandholesconfinedinInAsquantumdotmoleculeshaverecentlycome
5 toforeasapromisingsystemforthestorageormanipulationofquantuminformation. Wedescribe
1 here a feasible scheme for complete quantum optical control of two electron spin qubits in two
0 vertically-stacked singly-charged InAsquantumdots coupled by coherent electron tunneling. With
2 an applied magnetic field transverse to the growth direction, we construct a universal set of gates
thatcorrespondstothepossibleRamantransitionsbetweenthespinstates. Wedetailtheprocedure
n
to decompose a given two-qubit unitary operation, so as to realize it with a successive application
a
J of up to 8 of these gates. We give the pulse shapes for the laser pulses used to implement this
universal set of gates and demonstrate the realization of the two-qubit quantum Fourier transform
8
with fidelity of 0.881 and duration of 414 ps. Our proposal therefore offers an accessible path
to universal computation in quantum dot molecules and points to the advantages of using pulse
]
l shaping in coherent manipulation of optically active quantum dots to mitigate the negative effects
l
a of unintendeddynamics and spontaneous emission.
h
-
s
I. INTRODUCTION display bonding and anti-bonding states, that are sym-
e
m metric and anti-symmetric superpositions of localized
. Spins in semiconductor quantum dots1,2 (QD) are states.28 However, unlike natural diatomic molecules,
t which always have a bonding ground state, QDMs can
a promising candidates for satisfying the DiVincenzo
m criteria for quantum computation,3 such as state be tailored to have molecular ground states with anti-
- initialization,4–7 coherent spin manipulation,8,9 and bonding character.37
d qubit-specific measurements.10,11 The advantages of QD With the QDM embedded in a Schottky diode, the
n spin qubits over other qubit realizations are numerous. tuning of the voltage bias determines the stable ground
o Quantumdots arecompatible with existing semiconduc- state charge configuration.29 Especially important is the
c
[ tor technology and can be grown in millions in a reg- case of a doubly charged QDM,20,22,31,33 where each of
ular 2D array on a single millimeter-sized chip.12 QD the QDs is charged with a single electron or hole, since
1 spin qubits can be integrated on a chip with photonic thentheQDMcanbeusedasadoublespinqubitsystem.
v
crystal cavities,13–15 have short optical recombination An ambitious goal in quantum computation is complete
2
5 and photon emission times,16,17 can be manipulated by quantum control of such a multiple-qubit system. This
9 fastsingle-qubit8,18,19 andtwo-qubit20,21 quantumgates, control can be achieved by a universal set of quantum
1 and can be entangled with adjacent qubits by tunnel- gates, i.e. a set of gates with which an arbitrary uni-
0 ing interaction22 or with remote ones via entanglement tary operation can be constructed. Many universal sets
1. swappingwithphotons.23–25Recently,greatprogresswas of quantum gates were proposed. Ref. 38 showedthe set
0 made indevelopingmethods to increasethe QDspin de- of all two-level unitary gates is universal. Then, Ref. 39
5 coherencetimeeitherthroughspinecho26 orsuppression showedthissetofgatescanbeimplementedbycombina-
1 of nuclear-spin fluctuations.27 tions of single-qubit gates and the two-qubit controlled-
v: ThestudyoftheinteractionmechanismsbetweenQDs NOT (CNOT) gates. Other universal sets consisting of
i can lead to finding new ways to successfully manipu- only two-qubit gates were also found.40–43
X
late their quantum state, an important goal of quan- In this work, we present a scheme for complete quan-
r
a tum information technologies, like quantum computing tum control of a two-qubit system, a QDM composed
and quantum cryptography. Such coupling can be ob- of two vertically-stacked singly-charged InAs QDs sepa-
tained in optically active self-assembled QDs,28,29 with rated by a GaAs/AlGaAs tunnel barrier and embedded
techniques existing for ultrafast laser initialization,30 in a Schottky diode. The two electrons interact by ki-
measurement11,30andcoherentmanipulation20,22,31–33of netic exchange,which is based on coherent electron tun-
the spin qubits in these QDs. Many coupling mecha- neling and evidenced by a singlet-triplet splitting of the
nismsofopticallyactiveQDswerestudiedincludingelec- QDM energy levels. Quantum optical control is realized
trontunneling,22,32holetunneling,20,34exciton-mediated throughrealorvirtualexcitationofthebottomQDelec-
interaction31 and electron-hole exchange interaction.33 tron by laser pulses. The universal set of gates consists
In particular, it was found that embedding two coupled of the 5 two-level unitary operations that correspond to
QDs, which form a QD molecule (QDM), in a Schottky the possible Raman transitions in the Voigt geometry.
diode structure enables the tuning of the relative energy We find that every two-qubit unitary operation can be
levelsofthetwodots.35,36Likerealmolecules,QDMscan realized through at most 8 such Raman transitions, and
2
h d h
that pulse shaping the laser pulses can substantially in- B 0 T
crease the fidelity of the operation. As a case in point,
weshowareasonablyhighfidelity realizationofthe two-
U(z)
qubit quantum Fourier transform (QFT) in this system
V
taking light hole mixing, decay and decoherence into ac- 0
count. The results indicate the promise of QDMs as a
platform for quantum computation and the potential of
pulseshapingforincreasingthefidelityofquantumstate
manipulation in optically active QDs.
Thepaperisorganizedasfollows. InSec.IIwepresent
thesystemmodelandHamiltonian. WewritetheHamil-
tonian relative to an orthogonal basis, find the eigen-
V
statesandeigenenergies,andplottheallowedtransitions 0h
in the Faraday and Voigt geometries. In Sec. III we de-
scribe a universal set of gates for the QDM. We give an
algorithm for decomposing a given arbitrary two-qubit
unitaryoperationtoaproductofoperationsrealizedwith
U(z)
these gates and give criteria for a subset of these gates h
to still be universal. In Sec. IV we present the Hamilto- z
nianoftheinteractionwiththeelectromagneticfieldand
discussthe pulse shapesusedto implementRamantran-
sitions and the effects of light hole mixing. In Sec. V we FIG.1. Schematicdiagramofthebandstructureoftwover-
consider the effect of decay and decoherence. We write tically stacked self-assembled quantum dots embedded in a
the Lindblad equation for the system and derive the ex- Schottkydiode. Thegrowth direction isthez direction. The
pressionforthefidelityofanoperationrelativetoanideal height of the bottom/top dot is hB/hT, while the height of
one. InSec.VI weshow,forrealisticparameters,how to the interdot tunnel barrier is d0. Optical control of the two-
electron spin state is achieved through real or virtual transi-
applythe schemetorealizethe two-qubitQFT withrea-
tionstotheopticallyexcitedstatesusinglaserfieldstunedto
sonably high fidelity. Finally, in Sec. VII we discuss the
create an exciton at the bottom dot only. The electric field
key results and consider directions for future research. in the z direction, F,is exaggerated for clarity.
Hamiltonian for the ith electron, in turn, reads
II. SYSTEM MODEL AND HAMILTONIAN
p2 +p2 p2
h = i,x i,y + i,z +U(r ), (2)
i i
The system under consideration consists of two InAs 2m⊥ 2m(zi)
verticallystackedself-assembledquantumdotsseparated
where r and p (i = 1,2) are the position and momen-
by a GaAs/AlGaAs tunnel barrier and embedded in a i i
tum of the ith electron, and where the effective mass in
Schottky diode. The voltage of the diode is adjusted to
the z direction is m(z). The two-electron Hamiltonian
the chargestability regionwhere eachdot occupied with
is constructed by writing Eq. (2) for each electron and
a single electron at the ground state. The coherent tun-
adding the Coulomb interaction term. It is
nelcouplingoftheelectronsinthetwodotsismanifested
ininter-electronkineticexchangeinteraction,whichgives e2
rise to singlet and triplet electronstates delocalizedover H=h1+h2+ r 0r , (3)
both dots. The quantum states of the electrons are op- | 1− 2|
tically manipulated via laser fields that generate real or where e2 e2/(4πǫ )/ [ǫ(z ) + ǫ(z )]/2 , e being the
virtual electron-hole pairs (excitons) at the bottom dot. electron0ch≡arge,ǫ th0e v{acuu1m permi2ttivit}y andǫ(z) the
0
The magnetic field is initially taken as zero. relative dielectric constant.
With the growth direction taken to be in the z direc- Weconsiderthetwodotstobeseparatedsuchastojus-
tion,thepotentialexperiencedbyasingleelectroninthe tifyavariationaltreatmentintermsofatomic-likesingle-
conduction band is given by particle states localized at the individual dots. These
single particle states are B , B , T and
| i| ↑i | i| ↓i | i| ↑i
1 T ,where B /T is astatecorrespondingtothe so-
U(r)= 2m⊥ω⊥2(x2+y2)+U(z), (1) l|utii|o↓niof the S|chri¨od|iniger equation for the single-particle
Hamiltonian in Eq. (2) with U(z) containing only the
where m is the effective electron mass in the x and y bottom/top dot potential well. Listing the two-electron
⊥
directions, approximated as independent of z, ω is the combinations of these states gives the product state ba-
⊥
effective frequency of the parabolic confinement in the sis: , , , , , , , , ,0 and 0, ,where
|↑ ↑i |↑ ↓i |↓ ↑i |↓ ↓i |↓↑ i | ↓↑i
x and y directions, and U(z) is plotted in Fig. 1. The σ ,σ is the state with the bottom/top dot occupied
B T
| i
3
by an electron with spin projection σ /σ in the z di- new basis reads
B T
rection, 0 denotes an unoccupied dot, and denotes a
↓↑ + =2−1/2( ,0 + 0, ), (4)
doubly-occupied dot.
| i |↓↑ i | ↓↑i
S =2−1/2( , , ), (5)
| i |↑ ↓i−|↓ ↑i
T =2−1/2( , + , ), (6)
0
| i |↑ ↓i |↓ ↑i
T = , , (7)
+
| i |↑ ↑i
T = , , (8)
−
| i |↓ ↓i
=2−1/2( ,0 0, ), (9)
|−i |↓↑ i−| ↓↑i
where are the symmetric/antisymmetric doubly oc-
|±i
cupied states, S and T are the singlet and triplet
0
| i | i
Heitler-Londonstates,and T arethe twoothertriplet
±
| i
states. The spatial parts of the triplet states are anti-
symmetric, while those parts in the restof the states are
symmetric. This corresponds to the former/latter being
anti-bonding/bonding states.
With foresight, we choose a new basis composed of The Hamiltonian in Eq. (3) in the basis of Eqs. (4-9)
linearcombinationsoftheproductstatebasisstates. The is
UB+UT +J t +t 0 0 0 ǫ ǫ + UB−UT
2 B T B− T 2
t +t U +J 0 0 0 t t
B T BT B T
−
0 0 U J 0 0 0
H=(ǫB+ǫT)I + 0 0 BT0− UBT J 0 0 , (10)
b 0 0 0 0− UBT −J 0
ǫ ǫ + UB−UT t t 0 0 0 UB+UT J
B− T 2 B − T 2 −
where I is the identity matrix, and the following defini- write the Hamiltonian in Eq. (10) in the orthonormal
tions are employed: basis. The diagonalization of this Hamiltonian shows
b
the three triplet states T , T are degenerate eigen-
0 ±
ǫB = B hB , (11) states with an energy i|ndeipe|ndeint of the electric field
h | | i
ǫ = T hT , (12) F. The singlet state S is shifted in energy by the ki-
T
h | | i | i
netic exchange interaction below the triplet states, and
t= B hT , (13)
−h | | i this interaction also results in an admixture of the dou-
t =t u B T ǫ , (14)
B − BBBT −h | i B bly occupied states in the eigenstate dominated by
tT =t−uTTTB −hB|TiǫT, (15) |Si. Forthespecifice|±xpierimentalvaluesofhB =2.6nm,
J =uBTBT 2 B T t. (16) hT = 3.2 nm, d0 = 9 nm (see Fig. 1) and ∆S−T = 125
− h | i
µeV, taken from Ref. 22, we plot the eigenenergies of
Thesingle-particleHamiltonianhinEqs.(11-13)isgiven as a function of the external electric field F in Fig. H2.
in Eq. (2). The matrix elements of this Hamiltonian in The working point is taken as the one for which the
Eqs.(11-12)arethesingle-particleenergiesforthestates singlet-triplet splitting is ∆ = 125 µeV, and at this
S−T
B and T , apart from small contributions from the point the singlet eigenstate is numerically obtained as
| i | i
other quantum well in U(z). The matrix element t in S˜ =0.973S 0.173+ +0.149 . We termthisstate
Eq. (13) is termed the tunneling matrix element. The |togiether wit|hit−he tripl|etistates, t|h−eispin states.
Coulomb integral u (i,j,k,l=B or T) is given by
ijkl
Wenowconsidertheopticallyexcitedstatesinoursys-
u = dr dr e20 φ∗(r )φ∗(r )φ (r )φ (r ), tem, namely the X2− states which consist of a negative
ijkl Z 1 2 r r i 1 j 2 k 2 l 1 trion at the bottom dot and an unpaired electron at the
1 2
| − |
(17) top dot. These states are , , , , , ,
withφ (r)beingtheatomic-likesingle-particlewavefunc- and , , with |=↓↑⇑2−↑1i/2(| ↓↑⇑ ↓i | ↓↑⇓) ↑i
i
tionofthe state i . Forthe directCoulombintegralswe and | ↓↑⇓ ↓=i 2−1/2(| ↓↑⇑i ) | ↑re↓pir−ese|n↓ti↑nig|t⇑rii-
use the abbreviat|ioins U =u and U =u . ons, |an↓d↑⇓wiith =|3↑,↓3i−an|d↓↑i | =⇓i 3, 3 being the
ij ijji i iiii |⇑i |2 2i |⇓i |2 −2i
We make the basis of states in Eqs. (4-9) orthogonal heavy hole states with 3/2 and 3/2 spin projections
−
by replacing + and S in Eqs. (4-5) by (S + ). along z. The spin states can be excited to the optically
| i | i | i± | i
Wethennormalizethestatesoftheorthogonalbasisand excited states, which we term the trion states, through
4
(cid:112)(cid:110)(cid:158),(cid:110) (cid:112)(cid:110)(cid:158),(cid:112) (cid:112)(cid:110)(cid:160),(cid:110) (cid:112)(cid:110)(cid:160),(cid:112)
20
L
V 10
e
m
H E
E 0
tr
-
10
1 2 3 4 5 6 T T T
(cid:14) 0 (cid:16)
F HmV(cid:144)nmL
(cid:39)
S-T |S(cid:4)(cid:178)
FIG.2. Calculated spin stateenergy levels for thequantum
dot molecule system vs. F, the applied electric field in z di-
rection,withtheexperimentalparametersofofhB =2.6nm,
hT = 3.2 nm, d0 = 9 nm, ∆S−T =125 µeV and zero mag- FIG. 3. (color online) Energy level diagram of the quan-
netic field. The plotted energy levels at the working point tumdotmoleculeintheFaradaygeometrywithzeromagnetic
(F = 2.73 mV/nm), denoted by a dashed line, correspond, field. Thestatesincludethemodifiedsingletstate|S˜i;thede-
from bottom to top, to the modified singlet state |S˜i; the generate triplet states |T0i, |T+i and |T−i; and the optically
degenerate triplet states |T−i, |T0i and |T+i and two linear excited degenerate trion states | ↓↑⇑,↑i, | ↓↑⇑,↓i, | ↓↑⇓,↑i
combinations of the two doubly occupied states | ↓↑,0i and and |↓↑⇓,↓i with energy Etr. The kinetic exchange interac-
|0,↓↑iwithsmall(5.4% and0.05% attheworkingpoint)sin- tion shifts themodified singlet energy levelbelow that ofthe
glet state admixtures. The lowest/highest energy levels were triplet states with the splitting magnitude being ∆S−T. The
shifted down/up by 0.5 meV for the energy gaps to become selection rules are shown in the diagram with the red/blue
visible in the plot. These energy gaps are, from top to bot- arrows denoting σ± polarization.
tom, the anticrossing splitting of the doubly occupied states
whichequals35µeV andthesinglet-triplet splittingwhich is
35µeV at the center of the splitting (F =3.71 mV/nm) and Zeeman-splits the triplet states in energy to the states
125µeV at theworking point. T , T and T as shown in Fig. 4, while the
− x 0 x + x
| i | i | i
modifiedsingletstate,havingnospinprojectioninthe x
direction,is notshifted in energy. The new triplet states
the creationofanexcitonatthe bottomdotby the laser are given by
field. The4trionstateshavethesamesingle-particleand
T = , , (18)
Coulomb interaction terms in their energies. Moreover, − x
| i |− −i
we calculated for a wide range of relevant experimental T =2−1/2(+, + ,+ ), (19)
0 x
| i | −i |− i
values that the electron-hole spin exchange interaction
T = +,+ , (20)
+ x
is negligible, since the unpaired electron and hole reside | i | i
in different dots. We also found that hole tunneling is where
too weak to have a notable effect on the energies due to
the highly localized hole wave functions. We therefore =2−1/2( ) (21)
|±i |↑i±|↓i
conclude the trion states are degenerate.
are the eigenstates of the spin operator in x direction.
Withtheenergiesofthespinstatesandthetrionstates
The trion states are also Zeeman-split in energy, with
determined, we obtain the selection rules and plot the
the new states being t , and t , and where the
energy level diagram for the system in the Faraday ge- | + ±i | − ±i
states t are defined by
±
ometry and zero magnetic field in Fig. 3. The allowed | i
transitions are with circularly polarized light with po- t =2−1/2( ). (22)
±
larizations σ . We notice the T states are each iso- | i |↓↑⇑i±|↓↑⇓i
± ±
| i
latedfromthe restofthe spinstatesinthe sensethatno WeplotthetrionstatesZeemansplittingsinFig.4. With
Raman transition can be induced between each of these theenergylevelsdetermined,wewritetheselectionrules
statesandanotherspinstate. Thisresult,whichpersists for the Voigt geometry. We find the allowed transitions
when a magnetic field in the z direction is applied, pre- have linear polarizations in either the x or y directions,
cludes complete optical control of the two-electron spin andthatthesetoftransitions,showninFig.4,makesRa-
state in this geometry and, consequently ,we decide to man transitions possible between all pairs of spin states
consider the Voigt geometry with a magnetic field ap- apartfrom the pair of T and T , a point that will
− x + x
plied in the +x direction. be important later whe|n wie tack|le thie problem of com-
Theapplicationofamagneticfieldinthe+xdirection plete quantum control of the two-electron spin state.
5
t ,(cid:16) cation of a set of two-level unitary operations between
(cid:14) the spin states. Eachof the two-levelunitary operations
operates on two spin levels and is realized through two
(cid:39) t
h ,(cid:16) laserpulses withthe sameRamandetuning δ thatpump
(cid:16)
the transitions from the two spin states to a common
t ,(cid:14) (cid:39)(cid:14)(cid:39) trion state.
h
(cid:14) Supposewewanttorealizeagivenaunitaryoperation
U in the computational basis as a product of two-level
(cid:39)
h t ,(cid:14) unitary operations. A general proper unitary two-level
(cid:16) operation in the space of the states i and j (i,j =
| i | i
1,...,4) in the spin state basis , S˜ , T , T and
0 x + x
| i | i | i
T , is given by
− x
| i
θ
R (θ,n)=exp( i n σ), (23)
ij
− 2 ·
b b
which rotates the pseudo-spin vector in the subspace of
the states i and j by an angle θ about the axis n,
with σ =(σ| i,σ ,σ|)iacting in the subspaceof these two
x y z
T levels. Going back to the unitary operation U, we firbst
(cid:16) x write its matrix relative to the spin state basis. Then,
(cid:39) Ref. 38 shows how to decompose an arbitrary n-qubit
unitary operation to a product of up to 2n two-level
2
unitaryoperations. However,wepreferthe(cid:0)op(cid:1)erationsin
T (cid:39) the product to be proper unitary, as such operations are
0 x S-T more easily implemented experimentally,44 and we give
(cid:39) in Appendix A the procedure for decomposing U to a
S(cid:4)
product of up to 6 proper unitary two-level operators in
the form of Eq. (23) and an overall phase factor, which
T can be omitted.
(cid:14) x FromtheenergyleveldiagraminFig.4itcanbeenseen
thatapartfromR eachofthe 6two-levelunitaryoper-
34
atorsthatmayappearinthedecompositionofU,namely
FIG. 4. (color online) Energy level diagram of the quantum R , R , R , R , R and R , may be implemented
12 13 14 23 24 34
dot molecule in the Voigt geometry (growth direction is z,
with a single Raman transition. A graphical illustration
and magnetic field is in +x direction). The energy of the
of the situation appears in the graph in Fig. 5, wherein
modified singlet state |S˜i is unaffected by the magnetic field
each vertex represents a level and each edge a possible
andremainssplitfrom|T0ix by∆S−T,whilethetripletstates two-level unitary operation. The operation R may be
areZeeman-splitinenergyto|T+ix,|T0ixand|T−ix,withthe 34
implemented through 3 Raman transitions as shown by
splittingbeing∆=|ge|µBB,ge theelectrong-factor,µB the
Bohr magnetron and B the magnitude of the magnetic field. the following argument. If we represent a π rotation op-
Thedegeneracyoftheopticallyexcitedstatesisalsoremoved eration between the states i and j as
| i | i
bythemagneticfieldandthestatesareZeeman-splitasshown
inthefigure,where∆h =|gh|µBBandghistheholeg-factor. Pij =Rij(π,y), (24)
Theselectionrulesareplottedinthediagramwithredarrows
corresponding to V (πy) polarization, and blue arrow arrows then we have b
to H (πx) polarization.
R =P R P† , (25)
ik jk ij jk
R =P R P†. (26)
III. UNIVERSAL COMPUTATION WITH kj ik ij ik
TWO-LEVEL OPERATIONS
With the problem of R solved, we consider the fact
34
The spin states in the quantum dot molecule system that it may be advantageous to use of only some of the
in the Voigt geometry form a basis for a two-qubit com- two-level unitary operations, as the realizations of those
putational state space. This basis is , , where the using Raman transitionsmay be faster or havehigher fi-
|± ±i
states were defined in Eq. (21), and where the ad- delity relative to the other operations. We therefore ask
|±i
mixtures of the doubly-occupied states were omitted for whatuniversalsubsetsofthe5operationscanbechosen,
brevity. The scheme for universal computation we pro- i.e. subsets with which any arbitrary unitary operation
pose provides a realization of a given arbitrary unitary can be implemented. The answer to this question lies in
operation in the computational basis through the appli- thegraphinterpretationofEqs.(25-26). Theseequations
6
1 2 this result by induction, we find that an edge can be re-
alized by 2n 1 Raman transitions if a path of n edges
−
connects its vertices. An important result follows. For
a subset of the graph in Fig. 5 to be universal, it must
includeallverticesandbeconnected. Whenwedefineπ-
edges as edges representing π rotations, the graph for
±
theminimaluniversalsetofoperationscanbefoundand
its form is given in App. B.
IV. PULSE SHAPING
3 4
The two-levelunitary operations in our system are re-
alized through Raman transitions. For such a transi-
FIG. 5. Graph representation of the spin states and the tion to implement R the quantum dot molecule is il-
ij
possible two-level operations between them. The graph ver-
luminated by two phase-locked laser pulses propagating
ticesrepresentlevelsanditsedgesrepresentpossibletwo-level
in the z direction. The two pulses are linearly polar-
unitary operations. The vertices 1, 2, 3 and 4 are associated
with thestates |S˜i, |T0ix, |T+ix and |T−ix, respectively. ized, following the selection rules in Fig. 4, in either the
vertical (V) or horizontal (H) directions, have a com-
mon Raman detuning δ and pump the two transitions
from the spin states i and j to a common trion state.
| i | i
The model Hamiltonian in the basis of S˜ , T , T ,
0 x + x
| i | i | i
show that an edge may be realized by 3 Raman transi- T , t ,+ , t , , t ,+ , and t , , in the rotat-
− x + + − −
| i | i | −i | i | −i
tions if a two-edge path connects its vertices. Extending ing wave approximation (RWA) is
∆ 0 0 0 χ∗ χ∗ χ∗ χ∗
− 0S−T 0 0 0 χH∗ −χ∗V χ∗V −χH∗
V − H H − V
0 0 ∆ 0 √2χ∗ 0 √2χ∗ 0
0 0 −0 ∆ 0 V √2χ∗ 0 H √2χ∗
= H V ,
H χH χV √2χV 0 Etr +(∆h ∆)/2 0 0 0
−
χ χ 0 √2χ 0 E +(∆ +∆)/2 0 0
− V − H H tr h
χ χ √2χ 0 0 0 E (∆ +∆)/2 0
V H H tr− h
χ χ 0 √2χ 0 0 0 E (∆ ∆)/2
− H − V V tr− h−
(27)
where E is the trion state energy shown in Fig. 3, we g-factors,respectively. Theexpressionsχ (t)=Ω (t)/2
tr V V
have set h¯ = 1, ∆ = g µ B is the electron Zeeman andχ (t)=Ω (t)/2inEq.(27)are,respectively,halves
e B H H
| |
splitting, ∆ = g µ B is the hole Zeeman splitting, ofthe time-dependent Rabifrequency for the transitions
h h B
and where µ is t|he|Bohr magnetron, B is the magnetic S˜ t ,+ and T t ,+ .
B − 0 x −
| i→| i | i →| i
field magnitude, and g and g are the electron and hole In the interaction representation the Hamiltonian in
e h
Eq. (27) is recast as
0 0 0 0 χ∗ ei∆15t χ∗ei∆16t χ∗ei∆17t χ∗ ei∆18t
H − V V − H
0 0 0 0 χ∗Vei∆25t −χ∗Hei∆26t χ∗Hei∆27t −χ∗Vei∆28t
0 0 0 0 √2χ∗ei∆35t 0 √2χ∗ ei∆37t 0
V H
0 0 0 0 0 √2χ∗ ei∆46t 0 √2χ∗ei∆48t
= H V ,
H χHe−i∆15t χVe−i∆25t √2χVe−i∆35t 0 0 0 0 0
−χVe−i∆16t −χHe−i∆26t 0 √2χHe−i∆46t 0 0 0 0
χVe−i∆17t χHe−i∆27t √2χHe−i∆37t 0 0 0 0 0
−χHe−i∆18t −χVe−i∆28t 0 √2χVe−i∆48t 0 0 0 0
(28)
where ∆ = E E , and where E is the energy of approximated as the singlet state S . When the full
ij i j i
− | i
the state i . Eqs. (27) and (28) were derived with S˜
| i | i
7
expression for S˜ is used, the Hamiltonian matrices are where
| i
the same, apart from an extra numerical factor in the
off-diagonalmatrixelementsinvolving S˜ . Thisfactoris Iij =hi|U˜†Uid|ji, (38)
| i
the coefficient of S in S˜ , which for the experimental and where U˜ and U are, respectively, the actual and
| i | i id
values in Ref. 22 is 0.973.
ideal evolution operators. The expected fidelity in
As seen from the energy level diagram in Fig. 4, a
Eq. (37) is the yardstick by which we decide whether
Raman transition from one spin state to another may
one set of pulse parameters is better than another set of
suffer from unintended dynamics due to the presence of
parameters for the implementation of a given two-level
closeresonances. Ratherthanincreasethepulseslengths,
unitary operation R .
ij
which will make the operation slower and magnify the
We now incorporate the effects of light hole mixing in
effects of dephasing, we cope with the unintended dy-
our analysis. The hole states and in Eq. (22),
namics through pulse shaping,44,45 whereby we vary the denotedas H± ,andheretofor|e⇑taikenas|b⇓aireheavyhole
parameters of the pulses to optimize the operation. | z i
states, are better approximated through the Luttinger
We describe the explicit formofχV(t) andχH(t) with Hamiltonian47,48 as a superposition of heavy hole states
pulse shaping for the V-H, H-H and V-V cases, with the
and light hole states of the form
casenamesdenotingthepolarizationsoftheRamanlaser
pulses involved. In the V-H case we have H± =cosθ 3, 3 sinθ e∓iφm 3, 1 . (39)
| z i m|2 ±2i− m |2 ∓2i
χ (t)=χ (t)+χ˜ (t), (29)
V 0 2
InEq.(39)θ andφ arethemixingangles, 3, 3 are
χH(t)=χ1(t−∆t)+χ˜3(t−∆t), (30) the heavy homle statesm, 3, 1 the light hole s|t2at±es2,iand
|2 ±2i
where the quantization axis is taken as the growth direction (z
direction).
χ (t)=χ e−(t/τj)2e−iωjt+iϕj, (31)
j j WhenthenewholestatesinEq.(39)aresubstitutedin
χ˜j(t)=χj(t)eiϕ′jlncosh(t/τj), (32) thetrionstatesandtheHamiltonianmatrixiscalculated
following an analogous derivation in the single dot case
and where χj(t) (j = 0,1) is a main pulse with ampli- inRef.8,wefindresultsidenticalwiththeonesobtained
tude χj, pulse width τj, central frequency ωj and con- inthisreference,namelythatadjustingthepolarizations
stant phase ϕj. The shift in time between the two main ofthelasersfromV andHtoV′ =2−1/2(σ++eiµ+σ−)
pulses is given by ∆t. The expression χ˜j(t) (j = 2,3) and H′ =2−1/2(σ+ e−iµ−σ−), with
in Eq. (32) is a helping pulse designed to alleviate the −
effects of unintended dynamics. This pulse has an ad- eiµ± = √3cosθm±sinθme±iφm, (40)
ditionalchirpingterm,46 whichsweepsthe instantaneous √3cosθm sinθme∓iφm
frequencyintimethrougharangeoffrequenciescentered ±
the Hamiltonian has the same form as in the case with
about ω .
j
no light hole mixing [Eq. (28)] apart from replacing χ
The pulse shapes for the other two cases involve only V
and χ by
one polarization type. In the H-H case we have H
1+2cos2θ
χV(t)=0, (33) χ˜V′ =χV′ m , (41)
χ (t)=χ (t)+χ˜ (t)+ 3cosθm+√3e−iφmsinθm
H 0 2 1+2cos2θ
m
χ1(t−∆t)+χ˜3(t−∆t), (34) χ˜H′ =χH′3cosθm √3eiφmsinθm, (42)
−
and in the V-V case we have where χV′ and χH′ are the analogous quantities to χV
and χ for light with V′ and H′ polarizations, respec-
χ (t)=χ (t)+χ˜ (t)+ H
V 0 2 tively,andtheeffectsofV′ andH′ beingnon-orthogonal
χ (t ∆t)+χ˜ (t ∆t), (35)
1 3 are neglected.
− −
χH(t)=0. (36) Theidentityofthelastresultswiththe singledotcase
in Ref. 8 is not surprising, as only the bottom quantum
The pulse parameters should be varied so as to max-
dotisopticallyexcited. Tosummarize,theeffectsoflight
imize the fidelity of the operation relative to the ideal
holesmixingcanbecircumventedbyadjustingthepolar-
R . As this fidelity depends on the input state, the ex-
ij ization of the laser pulses and their Rabi frequencies as
pected value of the fidelity over all possible input states
noted above. With these adjustments, the Hamiltonian
is required. Let the states in the basis S˜ , T , T ,
| i | 0ix | +ix in Eq. (28) can still be used.
T , t ,+ , t , , t ,+ , and t , be denoted
− x + + − −
| i | i | −i | i | −i
by j (j = 1,...,8). For unitary evolution we find in
| i
App. C the expected fidelity is
V. DECAY AND DECOHERENCE
4 4
1
= (I I∗ + I 2), (37) When decay and decoherence are taken into account
F 20 ii jj | ij|
Xi=1Xj=1 the state of the system may be described by a density
8
matrix ρ and its non-unitary evolution by a quantum where ρ(ij) is the density matrix resulting from the non-
master equation in the Lindblad form unitary evolution of an initial density matrix i j . In
| ih |
the limit of unitary evolution (Γ = 0), ρ(ij) = U˜ i j U˜†
ρ˙ =−i[H,ρ]+ Li[ρ], (43) and Eq. (57) reduces to Eq. (37). | ih |
Xi
whereH isgiveninEq.(28),andwherethesumisoverall
trionstaterelaxationchannels,eachofwhichisdescribed VI. THE TWO-QUBIT QUANTUM FOURIER
by a Lindblad superoperator TRANSFORM
1 1
[ρ]=D ρD† D†D ρ ρD†D . (44)
Li i i − 2 i i − 2 i i As a case in point of a non-trivial two-qubit transfor-
mation realized using the methods described above, we
In taking the relaxation channels as separate and inco-
choose the two-qubit quantum Fourier transformation.
herent, we neglected spontaneously generated coherence
Inthe computationalbasis , this transformationis
(SGC),sinceweassumedthe Zeemansplittingsarelarge |± ±i
enough to satisfy ∆ Γ (j,k = 5,...,8; j = k),
| jk| ≫ 6 1 1 1 1
where Γ is the total relaxation rate of a given trion
1 1 i 1 i
level,49,50 and that the level splittings ∆, ∆h and ∆S−T UQFT = 2 1 1 −1 −1. (58)
are larger or of the same order as the Rabi frequency 1 −i 1 −i
Ω.51 The effects of pure dephasing are not included in − −
Eq. (43), as this process, which is generated by the nu-
Rewriting Eq. (58) relative to the physical basis S˜ ,
clear spins, has a rate much lower than the total trion | i
dephasing rate in this system.9,52 T0 x, T+ x, and T− x as UQFT, andapplying the algo-
| i | i | i
rithm inApp. A to decomposethe resultto a productof
The 12 jump operators, D , in Eq. (43), each corre-
i e
two-level proper unitary operations, we find
sponding to a possible relaxation channel, are
D = Γ/4S˜ t ,+, (45) U =eiαR (θ ,n )R (θ ,n )R (θ ,n )
1 | ih + | QFT 32 32 32 34 34 34 12 12 12 ×
p
D2 = Γ/4|T0ixht+,+|, (46) e R14(θ14,n14b)R24(θ24,n24b), b (59)
p
D3 = Γ/2|T+ixht+,+|, (47) where Rij(θ,n) is dbefined in Eqb. (23), and where α, θij
D4 =pΓ/4S˜ t+, , (48) and nij were numerically determined.
p | ih −| We detail tbhe optimization process of the pulse train.
D5 = Γ/4|T0ixht+,−|, (49) Firstb, we choose the physical parameters for the system
p
D = Γ/2T t , , (50) as the ones in Ref. 22, namely h = 2.6 nm, h = 3.2
6 − x + B T
| i h −|
D =pΓ/4S˜ t ,+, (51) nm, d0 = 9 nm (see Fig. 1) and ∆S−T = 125 µeV.
7 | ih − | The electron and hole g-factors are taken from Ref. 6 as
p
D8 = Γ/4|T0ixht−,+|, (52) ge =−0.48andgh =0.31,andthe trionstateslinewidth
p Γ is obtained from Ref. 4 as Γ=1.2 µeV. Next, for each
D = Γ/2T t ,+, (53)
9 + x −
| i h | of the operations R , R , R , R and R we choose
D10 =pΓ/4S˜ t−, , (54) the trion state that12will13be u1s4ed t2o3 realize24the Raman
| ih −|
D =pΓ/4T t , , (55) transition. Wechoosethisstatesoastoreducetheunin-
11 0 x −
p | i h −| tendeddynamics,i.e. suchthattheresonancefrequencies
D = Γ/2T t , . (56)
12 | −ixh − −| ofeachofthe transitionsfromthe twospinstatesto this
p
The terms in the square roots in Eqs. (45-56) are the trion state is as far as possible from other allowed tran-
relaxation rates for the corresponding decay channels. sitions with the same polarization. We then choose a
These rates were determined by splitting the totaldecay value for the magnetic field B that maximizes the min-
rate, the trion state linewidth Γ, in the ratio of absolute imum absolute frequency difference among all pairs of
square values of the dipole matrix elements for the al- allowedtransitionswiththesamepolarization. Formag-
lowed transitions from the this state to the spin states, netic field values lower than 10 T, this value is found to
asthesevaluesareproportionaltothespontaneousdecay be B = 8.99 T, and the minimum absolute frequency
rates of the corresponding channels. difference is 36.5 µeV.
With spontaneous decay and decoherence included in We then turn to the optimization of the individual
the model, the expression for the expected fidelity of an pulses. SinceeachR inEq.(59)(apartfromR )canbe
ij 34
operation relative an ideal operation in Eq. (37) should implemented directly as given or indirectly via Eqs. (25-
be generalized. The derivation in App. D gives this gen- 26),weconsiderallpossibleindirectimplementationsand
eralized expected fidelity as optimize the 35 operations R (θ ,n ), R (θ ,n ),
ij 32 32 ij 34 34
= 1 4 4 j U†ρ(ji)U i + j U†ρ(ii)U j , Rwhije(rθe12(,i,nj1)2)=, (R1i,j2()θ,1(41,,n31)4,)(,1,R4i)j,((θ22,43,)nb,2(42),,4)P.iIjnaenadchbPoip†j-,
F 20 (cid:20)h | id id| i h | id id| i(cid:21) b b b
Xi=1Xj=1 timizationwerealizetheoperationwithtwomainpulses,
(57) eachofwhichhavingthe formofEq.(31),andmaximize
9
the expected fidelity in Eq. (37) through grid search in Our results indicate that pulse shaping can substan-
pulse parameter space followed by gradient descent. In tiallyincreasethe fidelity ofquantumoperationsinopti-
the optimization process we keep in mind the need for callyactiveQDs,andthatthe QDMisa promisingplat-
short pulses that will ensure fast operation. Examining form for multi-qubit quantum computation. The main
the results of the optimizations, we conclude that direct limitations for increasing the overall fidelity of a given
implementation is always preferable when possible, with two-qubitoperationwiththisschemeareunintendeddy-
the only case where indirect implementation is chosen namics, decay and decoherence. The first can be dealt
being with through improved pulse shaping. Due to limited
computing power, we chose our pulse search algorithm
R34(θ34,n34)=P13R14(θ34,n34)P1†3, (60) to optimize eachpulse separately. A better optimization
maybeachievedthroughsimultaneousoptimizationofall
which cannot be dbirectly implementedbdue to the selec-
4 pulses ofa givenoperation. Amelioratingthe effects of
tion rules.
decoherenceanddecaycanalsobe achievedbyimproved
With the pulse train determined to consist of 7 oper-
pulse search methods. In our search we maximized the
ations, we add two helping pulses, each in the form of
fidelity expression in Eq. (37) that does not take decay
Eq. (32), to each operation. The parameters of the first
anddecoherenceintoaccount,ratherthantheexpression
helping pulse are varied so as to maximize the fidelity
that takes these effects into account in Eq. (57). Using
in Eq. (37) and the process is repeated for the second
the latter expression, which takes much longer to evalu-
helping pulse. Following the optimization of the helping
ate, will increase the fidelity of the overalloperation.
pulses, we turn to the optimization of the time spacings
Theexperimentalrealizationofuniversalcomputation
between the 7 operations. The fidelity we maximize is
in QDMs will be a major step towards building a QD-
now the fidelity of the entire QFT with decay and de-
basedoptically-controlledquantumcomputer. Futurere-
coherence taken into account as given in Eq. (57). The
search may look into methods for robust and effective
resultingoptimizedpulsetrainsarecalculatedtohavean
entanglement generation between distant QDMs, possi-
expectedfidelity of84.9%anda durationof453pswhen
bly with multi-photon light, as was suggested for single
helpingpulsesarenotemployed,andanexpectedfidelity
QDs.53,54 Quantum computations with more than two
of 88.1% and a duration of 414 ps when helping pulses
qubits may be realized in a quantum network of entan-
areused. The mainsourceofthe infidelity is unintended
gled QDMs through unitary operations on each of the
dynamics,thatis excitationoflevelsother thanthe ones
QDMs,possiblyemployingthetheoryofclusterstates.55
intended. This result is revealed comparing the fideli-
ties in Eq. (37) and Eq. (57) for each of the individual
operations. We find the decrease in fidelity from unity
Appendix A: Decomposition of a Unitary Operation
due to decoherence and decay is, on average, 0.8% per
to a Product of Two-Level Proper Unitary
operation, while the decrease due to unintended dynam- Operations
ics is 2.6% per operation. This result could have been
expected as the minimum absolute frequency difference
In this appendix we show a procedure to decompose
of 36.5 µeV is always comparable to or smaller than the
a given unitary operation U to a product of an overall
spectral width of the pulses.
phasefactoreiα andtwo-levelproperunitaryoperations.
Theprocedureisgivenforthecaseofa3-statebasis,but
can be easily extended to an N-state basis with N >3.
VII. CONCLUSIONS
Thefirststepinthealgorithmistofactorout U =eiα
andproceedwithU′ =e−iα/3U,whichisproper| u|nitary.
In this work, we demonstrated a scheme for complete Let U′ be given by
quantum control of a system of two singly-charged QDs
coupled by coherent electron tunneling. We derived the a d g
energy level diagram for the system and chose the two- U′ = b e h . (A1)
leveloperationsrealizedby the 5 possible Ramantransi- c f g
tionsthroughthetrionstatesastheuniversalsetofgates.
If b = 0, we take U as the identity matrix and proceed
In contrast with the commonly used universal computa- 1
to the next step. Otherwise we take U as the proper
tion scheme of CNOT gates and single-qubit gates,39 we 1
unitary matrix
used only two-qubit gates, each realizable through a few
laser pulses. Our choice is advantageous as single-qubit a∗ b∗ 0
1
gates, which can be described as two simultaneous two- U1 = b a 0 (A2)
qubit gates, tend to suffer from unintended dynamics in a2+ b2 −0 0 1
| | | |
this system.22 We demonstrated our scheme by describ- p
and have
ing the realization of the two-qubit QFT in the QDM
system with a reasonably high fidelity of 88.1%. We in- a′ d′ g′
corporated light hole mixing, decay and decoherence in U1U′ = 0 e′ h′ . (A3)
our analysis. c′ f′ j′
10
1 2
Then, if c′ = 0, we take U as the identity matrix and
2
proceed to the next step, while if c′ =0, we define
6
a′∗ 0 c′∗
1
U2 = 0 1 0 , (A4)
|a′|2+|c′|2 c′ 0 a′ (cid:652) (cid:652)
p −
which is proper unitary, and have
1 0 0
U2U1U′ =0 e′′ h′′ U3. (A5)
0 f′′ j′′ ≡ 3 4
Hence, the sought decomposition is
U = U 1/3U−1U−1U . (A6)
| | 1 2 3 FIG. 6. Graph representation of the minimal subset of the
possible two-level operations that can still achieve universal
computation. Thegraphverticesrepresentlevelsanditsedges
Appendix B: The Minimal Universal Set of
representpossibletwo-levelunitaryoperations. Aπ-edgerep-
Operations in the Quantum Dot Molecule
resents a ±π rotation operation. The vertices 1, 2, 3 and 4
are associated with the states |S˜i, |T0ix, |T+ix and |T−ix,
Inthisappendix,weprovetheminimaluniversalsetof respectively.
two-levelunitaryoperationsinthequantumdotmolecule
system is represented by the graph in Fig. 6. In the last
paragraph of Sec. III it was shown that a graph repre- while the output state is
senting a universal set of operations must be connected
and containallvertices. Hence, the minimum number of 8
edgesis3. Theedgescannotallbeπedgessincethenthe ψout =U˜ ψin = cj j , (C2)
| i | i | i
number of possible two-qubit unitary operations attain- Xj=1
able with a finite number of such two-level operations is
finite. We conclude the set of operations represented by with U˜ being the evolution operator. The ideal output
the graph in Fig. 6 is minimal. state is given by
This set of operations is universal, since each of the
4
6 two-level unitary operations that can appear in the
ψ =U ψ = d j , (C3)
decomposition of an arbitrary unitary operation U can | idi id| ini j| i
bewrittenintermsofthe3operationsinthesetthrough Xj=1
use of Eqs. (25-26), namely
where U is the ideal evolution operator. The fidelity
id
R =R , (B1) of the output state relative to the ideal output state is
12 12
calculated as
R =P R P† , (B2)
13 23 12 23
R =P P R P† P† , (B3) = ψ Tr (ψ ψ )ψ , (C4)
14 21 14 21 14 21 F h id| T | outih out| | idi
R =P P R P† P† , (B4)
23 12 23 12 23 12 where we trace overthe trion states. PluggingEqs.(C2)
R =P R P† , (B5) and (C3) in Eq. (C4), we find
24 14 21 14
R =P P R P† P† , (B6)
34 14 23 21 23 14 2
4
wspiethctiRveijly.and Pij defined in Eq. (23) and Eq. (24), re- F =(cid:12)(cid:12) c∗jdj(cid:12)(cid:12) . (C5)
(cid:12)Xj=1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Averaging Eq. (C5) over all possible input states as in
Appendix C: The Expected Fidelity for Unitary
Ref.44,theexpectedfidelityoftheoperationisobtained
Evolution
in the form of Eq. (37).
In this appendix, we derive the expression for the ex-
pected fidelity of an operation relative to an ideal oper- Appendix D: The Expected Fidelity for
ation when state evolution is unitary. The input state Non-Unitary Evolution
ψ is given by
in
| i
4 In this appendix, we derive the expected fidelity of
ψ = b j , (C1) an operation relative to an ideal operation when state
in j
| i | i
Xj=1 evolution is non-unitary. Let the initial state be a pure
