Table Of ContentTime evolution and decoherence of entangled states realized in coupled
superconducting flux qubits
Takuya Mouri1,2, Hayato Nakano2,3, and Hideaki Takayanagi1,2,3
1Department of Physics, Tokyo University of Science, Tokyo 162-8601, Japan
2CREST, Japan Science Technology Agency, Saitama 332-0012, Japan
5 3NTT Basic Research Laboratories, NTT Corporation, Kanagawa 243-0198, Japan
0
(Dated: February 2, 2008)
0
2 Westudytheoreticallyhowdecoherenceaffectssuperpositionstatescomposedofentangledstates
n ininductivelycoupledtwosuperconductingflux-qubits. Wediscoverthatthequantumfluctuationof
a anobservableinacoupledflux-qubitsystemplaysacrucialroleindecoherencewhentheexpectation
J value of the observable is zero. This examplifies that decoherence can be also induced through a
8 quantummechanicallyhigher-ordereffect. Wealsofindthatthereexistsadecoherencefreesubspace
2 for theenvironment coupled via a charge degree of freedom of the qubitsystem.
] PACSnumbers: 03.67.Lx,74.50.+r,85.25.Cp
l
l
a
h I. INTRODUCTION whereγ (i=L1,R1,L2,R2,L3,R3,M)isthephasedif-
i
-
ferenceacrosseachjunctionortheinductance. Bycalcu-
s
e Josephson-junctioncircuits behave quantum mechani- lating the Hamiltonian with parameters EJ = 100GHz,
m cally,iftheyaresufficientlydecoupledfromtheirenviron- E = (2e)2 = 8GHz, α = 0.75, M = 1 (h¯ )2, we can
C 2C 20EJ 2e
t. ment. However, in reality, since a macroscopic quantum obtain the energy dispersion shown in Fig. 1(b) and we
a state is difficult to be decoupled from their environment call the point at f =0.5 as the “degeneracy point”.
m
completely, a quantum mechanical superposition state
- suffers decoherence. This is one of the central problems
d
that must be solvedbefore these circuits can be used for
n
o quantuminformationprocessing[1]. Inordertominimize
c decoherence on quantum states, it is very important to
[ search what is the dominant source of decoherence and
how sensitively qubits feel the effect of environment. In
2
v this paper, we theoretically present that the quantum
1 fluctuationofanobservableinacoupledflux-qubitplays
8 acrucialroleindecoherence,eveniftheexpectationvalue
5 ofthe obseravableisalwayszero. We alsoreportthatwe FIG. 1: (a)Effective circuit of inductively coupled two flux
1 qubits. Two superconducting closed loops correspond to two
canextractadecoherence-freesingle-qubitbasisfroman
0 flux qubits. A qubit has three Josephson junctions. An ex-
inductively coupled flux-qubit system.
5 tenal flux Φ1 or Φ2 is penetrating through each qubit loop.
0 Qubits are coupled to each other through the mutual induc-
/ tance M. (b)Energy dispersions obtained when identical ex-
t
a II. INDUCTIVELY COUPLED TWO FLUX ternal fluxes are applied to the qubits, f = Φ1/Φ0 = Φ2/Φ0
m QUBIT SYSTEM (Φ0 =h/2e).
-
d Among qubits based on Josephson-junction circuits, a WecanapproximatetheHamiltonianforthetwo-qubit
n
flux qubit was realized as a superconducting ring with system using Pauli matrices σz, σx,
o i i
three Josephson-junctions[2,3] and coherent oscillations
c
: were demonstrated[4,5,6]. We discuss decoherence ap- HS =h1σ1z +∆1σ1x+h2σ2z +∆2σ2x+jσ1zσ2z.
v
pearing in the dynamics of an inductively coupled two
i Here, subscripts 1 and 2 indicate the left and right
X superconducting flux-qubits in Fig. 1(a)[7,8]. Four of
qubits, respectively. The basis ofthe two-qubitstate are
Josephson-junctions have the Josephson energy E and
r J
, , and ,where corresponds
a capacitance C. The remaining two have αEJ and αC, |↓1↓2i|↓1↑2i|↑1↓2i |↑1↑2i |↑ii
to the state where the persistent current flows counter-
where α = 0.75 is a constant. The Hamiltonian of the
clockwiseand totheclockwisestatearoundtheloop
system is given by |↓ii
of the qubit i(i = 1,2). h (i = 1,2) is the parameter
i
C ¯h 2 that depends on the externalflux Φi for the qubit i, and
H = γ˙2 +γ˙2 +γ˙2 +γ˙2 +α(γ˙2 +γ˙2 ) h =I (Φ Φ0)= I Φ (f 1), where I (i= 1,2) is
2 (cid:18)2e(cid:19) L1 R1 L2 R2 L3 R3 i pi i− 2 pi 0 i− 2 pi
(cid:0) (cid:1) the absolute value of the persistent current in the each
−EJ(cos[γL1]+cos[γR1]+cos[γL2]+cos[γR2](1) qubit determined by EJ and α. Φ0 = h/2e is the flux
1 ¯h 2 quantum. ∆i(i = 1,2) is the tunneling matrix element
+α(cos[γL3]+cos[γR3]))+ 2M (cid:18)2e(cid:19) γM2 , eachqubit has, which is the order of √ECEJe−√EJ/EC.
2
j is the coupling constant between two qubits, which is tion[9],
determined approximately by MI I . If the external
p1 p2
flux is equally applied (h =h =h) and the two qubits 1 t
1 2 ρ˙ = dt (ν(t )[x,[x( t ),ρ ]]
are assumed to be identical (∆1 = ∆2 = ∆), we can S ¯hZ0 1 1 − 1 S
simplify the Hamiltonian as
iη(t )[x, x( t ),ρ ]),
1 1 S
− { − }
HS =hσ1z +∆σ1x+hσ2z+∆σ2z +jσ1zσ2z. where
By comparing energy eigenstates of the H and Fig. ∞ ω
S ν(t)= dωcos[ωt]coth[ ]J(ω),
1(b)nearthedegeneracypoint,wefindthatthisapprox- Z0 2kBT
imatedHamiltonianH reproduceswellthe resultofthe
S ∞
η(t)= dωsin[ωt]J(ω).
original two flux-qubit Hamiltonian with j = 1.78GHz
Z
0
and ∆ = 2.07GHz. Therefore, hereafter we analyze the
decoherence in our coupled flux-qubit system with this k is the Boltzman constant and T is the absolute tem-
B
HspSo.ndIns tteormthseoflfutxhethoeriqguinbailtfliuhxa-sq,uabnitdsσyixstteomt,hσeizcchoarrrgee- pHeerraetuaries.tWheecaudtoopfft tfhreeqsupeencctyraalnddenwseityseJt(aω=)=5.ga2ω+aω22.
polarization in the junction of the qubit. At the degen-
eracy point h = 0, we can obtain eigenvalues and eigen-
states[7], IV. RESULTS AND DISCUSSION
E0 = j2+4∆2, ψ0 = 1 2 + 1 2 A. decoherence via flux degree
− | i |↑ ↑ i |↓ ↓ i
p
j+ j2+4∆2
( )( + ),
− p2∆ |↑1↓2i |↓1↑2i First, we assume thatthe interactionbetweenthe sys-
E = j, ψ = , temandtheenvironmentiscausedonlythroughthetotal
1 1 1 2 1 2
− | i |↑ ↓ i−|↓ ↑ i flux of qubits; then we take x = σz +σz. This is a rea-
E =j, ψ = , 1 2
2 | 2i |↑1↑2i−|↓1↓2i sonable situationbecause we canexpect that the system
E = j2+4∆2, ψ = + (2) suffers decoherence due to the coupling to the detector
3 3 1 2 1 2
| i |↑ ↑ i |↓ ↓ i
p like a SQUID and the fluctuation of the external flux,
j j2+4∆2
−( −p2∆ )(|↑1↓2i+|↓1↑2i). both of which are related to small flux generated by the
persistent currents flowing in qubits. Now, we consider
Here, a remarkable fact is that all the eigenstates of a superposition state composed of the first and second
the two-qubit Hamiltonian at the degeneracy point are excited states |ψ12(0)i = √12(|ψ1i+|ψ2i) at the degen-
strongly entangled ones. Adopting a state composed of eracy point(h = 0) as the initial state. We can analyti-
these entangled states as an initial state, we will dis- cally calculate the time evolutions of expectation values
cussthe time evolutionofthe density operatorandsome σz , σz , σx and σx when there is no decoherence.
h 1i h 2i h 1i h 2i
quantities. Provided that it is possible to set independent probes
whichcandetect both flux andchargeoneachofqubits,
we can measure expectation values σz , σz , σx and
III. PHYSICAL MODEL OF DECOHERENCE hσ2xi independently. |ψ12(t)i and thhe 1tiimeh e2violhut1ioins of
AND QUANTUM MASTER EQUATION observables at h=0 are obtained as follows;
In order to take into account decoherence, we con-
ψ (t) =cos[jt] +isin[jt] , (3)
sider the total system as the sum of the qubit system | 12 i |↓x1↑x2i |↑x1↓x2i
ψ (t)σz ψ (t) =0,
and the environment. The total Hamiltonian is HTOT = h 12 | 1| 12 i
H + H + H , where H , H , and H , are the ψ (t)σz ψ (t) =0,
S E INT S E INT h 12 | 2| 12 i
Hamiltonian of the qubit system, the environment, and ψ (t)σx ψ (t) = cos[2jt],
the interaction between them, respectively. We showed h 12 | 1| 12 i −
ψ (t)σx ψ (t) =cos[2jt].
already HS and the components of environment and in- h 12 | 2| 12 i
teraction are described as follows; H = ¯hω a a ,
E n n †n n Here xi = i + i , xi = i + i
HINT = x kλk(ak +a†k). We regard thePenvironment (i = 1|,2↑).iThis| s↑hoiws |th↓atith|is↓coiheren−t |os↑ciillatio|n↓isi
as a bosonPbath. ak, a†k arethe annihilationand the cre- impossible to detect through the flux, but there is a
ation operators of the environmental mode k. λ is the possiblity to detect through the charge. We will show
k
strengthofthecouplingineachmode. xmeansaphysical the result of the numerical calculation in the presence of
quantity of the qubit system that is coupled to the en- decoherence, which is obtained by solving the quantum
vironment. Assuming that the system-environment cou- masterequationwiththe aboveinitial state. As wehave
pling is small enough, we can treat H as a pertur- expected in the analytical calculation, we realize that
INT
bative term so that we acquire a quantum master equa- expectation values σz and σz obtained in numerical
h 1i h 2i
3
calculations are always zero. The most interesting fea- decoherence via x = σz +σz. We can again observe the
1 2
ture of this example is that the expectation value of the expectation value of the coupling between the qubit sys-
physical quantity which interacts with the environment tem and the environmentis always zero ( σz+σz =0).
σz+σz iszero. Althoughwewouldspeculatethatthere Fromtheabovetwoexamples,surelywechan1mak2eicoher-
h 1 2i
is no coupling between qubits and the environment, this ent oscillations where hHINTi = hxi kλk(ak +a†k) = 0
turns out not true according to the calculation results. by using initial states which is comPposed by entangled
Figure 2(a) shows the oscillations of σx and σx . We energy eigenstates. However, one is strongly affected by
h 1i h 2i
decoherenceandtheotherisnot. Then,weareinterested
in considering how quantum fluctuations of the coupling
x may become the source of decoherence[8]. In fact, the
calculations obtaining Figs. 2 and 3 also give the finite
quantum fluctuations
ψ (t)δx2 ψ (t) 1.4,
12 12
h | | i∼
p
ψ (t)δx2 ψ (t) 0.75,
01 01
h | | i∼
p
FIG. 2: (a)Time evolution of expectation values hσ1xi and where δx2 x2 x 2. It looks like that the quantum
x ≡ − h i
hσ2iinthepresenceofdecoherence. (b)Timeevolutionofthe fluctuationsofthecouplingxandresultingmagnitudeof
eigenvalues pi(i = 1,2,3,4) of the density operator. (h = 0, decoherence are determined by the curvatures of energy
j = 1.78, ∆ = 2.07, g = 1) Time is normalized by ∆. The dispersions in Fig. 1(b) around the degeneracy point
initial state is |ψ12(0)i. Decoherence is caused through x = f = 0.5. The quantum fluctuation of x = σz + σz
σ1z+σ2z. qualitatively corresponds to the quantum fluctua1tion o2f
an applied flux in the horizontal axis of Fig. 1(b). In
find that thier amplitudes are obviously suffering deco-
fact, by a perturbation calculation under the condition
herence. Time evolution of eigenvalues of the density
h ∆, j , the energy dispersion of an eigenstate i
operator in Fig. 2(b) helps us realize that a pure state | | ≪ | | | | | i
in the vicinity of the degeneracy point is given by
is rapidly changing into a mixed state due to decoher-
encebecausetheemergenceofthefinitesecondandlater E E0 h2 hi0|σ1z +σ2z|j0ihj0|σ1z +σ2z|i0i,
eigenvalues means the system becomes a mixed state. i∼ i − E0 E0
Xj=i j − i
Therefore, we can make sure that, even if the expecta- 6
tionvalueoftheinteractingquantitybetweensystemand whereE0and i0 aretheeigenenergyandeigenstateat
environment is always zero, there exists certain effect of the degeneiracy p|oiint. So, the curvature ∂2Ei(i=0,1,2)
decoherence. ∂h2
is approximately proportional to
Next, we discuss when the initial state is the super-
position state ψ (0) = 1 (ψ + ψ ) of the ground i0σz +σz j0 j0σz +σz i0 ,
| 01 i √2 | 0i | 1i h | 1 2| ih | 1 2| i
and the first excited states. Figure 3(a) shows oscilla- Xj
thatisthequantumfluctuationofσz+σz. Asaresult,
1 2
wehavereachedaconclusionthatthedecoherenceatthe
degeneracypointdependsonthecurvatureoftheenergy
dispersion around the degeneracy point.
Now we discuss the concurrence which is defined as
C(t)=√ξ √ξ √ξ √ξ whereξ (i=1,2,3,4)isthe
1 2 3 4 i
− − −
i theigenvalueofΛ(t) p 2 ψ σyσy ψ 2. This is
≡ i| i| |h i| 1 2| ii|
a very important measurPe because the concurrence tells
us how the two qubits keep entanglement which provide
FIG. 3: (a)Time evolution of expectation values hσ1zi and us the possiblity to implement quantum parallelism cal-
hσ2zi with decoherence via x = σ1z +σ2z. (b)Time evolution culations. Figure 4 shows the time evolution of the con-
of eigenvalues of the density operator. The initial state is currence. When compared with the decays of the corre-
|ψ01(0)i. sponding coherentoscillations in Figs. 2(a) and 4(a), we
canfindthattheentanglementdecaysmorerapidlythan
tions of the flux induced at the qubit1 and qubit2, and the observables σx and σx in the presence of decoher-
h 1i h 2i
these oscillations suggest that the possibility to detect ence.
the fluxes σz , σz separately by setting an indepen-
h 1i h 2i
dent SQUIDs over each of the qubits. Figure 3(a) also
shows that amplitudes of oscillations hardly decay and B. decoherence via charge degree
it seems that the oscillations are hardly affected by de-
coherence. This suggests that it could be possible to Next,wesupposethattheinteractionbetweenthesys-
realize coherent oscillation which is robust against the temandtheenvironmentisonlythroughthetotalcharge
4
correction codes[12]. Figure 5(b) represents time evolu-
tion of the concurrence and we can make sure that the
entanglement never deteriorates because of the suppre-
sion of decoherence in the DFS.
FIG.4: (a)Timeevolution oftheconcurrencewiththeinitial
state|ψ12(0)i. (b)Timeevolutionoftheconcurrencewiththe
initial state |ψ01(0)i.
freedomthe qubits have; x=σx+σx. For example, this
1 2 x
typeofinteractionmightappearinthepresenceofresid- FIG. 5: (a)Time evolution of the expectation values hσ1i,
x
ual charges near the qubit junctions. When the initial hσ2i. (b)Time evolution of concurrence. The initial state is
x x
state is ψ (0) = 1 (ψ + ψ ), as we have analyti- |ψ12(0)i. Decoherence is caused through x=σ1 +σ2.
| 12 i √2 | 1i | 2i
callycalculated,wehavealreadyknown σz and σz are
alwayszeroand ψ (t)σx ψ (t) and ψh 1i(t)σxhψ2i(t) V. SUMMARY AND CONCLUSION
h 12 | 1| 12 i h 12 | 2| 12 i
oscillate sinusoidally. The numerical calculation of the
time evolution gives the behavior of the expectation val- In summary, we analyzed the time evolution of an in-
ues, shown in Fig. 5(a). We can see hσ1xi and hσ2xi ductively coupled flux qubit system in the presence of
amazingly keep oscillating without any reduction of the environment. We confirmed the importance of quantum
amplitudes, and we have confirmed that the pure state fluctuations of the observable coupled to the environ-
is kept perfectly. We find this is because the two basis ment. WealsofoundtwobasisstatesconstructingaDFS
states ψ1 and ψ2 composing the initial state ψ12(0) for charge fluctuations.
| i | i | i
constructa Decoherence Free Subspace(DFS)[10] for the
coupling between the two qubit system and the environ-
ment via x = σx +σx[10,11]. In other words, ψ and
1 2 | 1i Acknowledgements
ψ are the degenerate eigenstates of x. Therefore, we
2
| i
cansuggestthatitispossibletodesignasinglequantum
bit which does not form entanglement with the environ- WethankF.Wilhelmforvaluablediscussions. Wealso
ment through the charge degree of freedom for a super- acknowledge K. Semba, S. Saito, H. Tanaka, F. Deppe
position state of ψ and ψ . This is an example of a andJ.Johanssonfor their comments fromthe viewpoint
1 2
| i | i
suppression of decoherence by introducing a redundant ofexperiments. M. Ueda shouldbe acknowledgedforhis
qubit and a qubit-qubit interaction, like quantum error helpful advices.
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