Statistics and noise in a quantum measurement process

4 Pages·0.36 MB·English

by  Yuriy Makhlin

Statistics and noise in a quantum measurement process
Yuriy Makhlin1,2, Gerd Scho¨n1,3, and Alexander Shnirman1
1Institut fu¨r Theoretische Festk¨orperphysik, Universit¨at Karlsruhe, D-76128 Karlsruhe, Germany
2Landau Institute for Theoretical Physics, Kosygin st. 2, 117940 Moscow, Russia
3Forschungszentrum Karlsruhe, Institut fu¨r Nanotechnologie, D-76021 Karlsruhe, Germany
0
0
0
2 the initial state of the qubit. The origin of the mix-
The quantum measurement process by a single-electron
ing is that the charge operator (the measured quantity)
n transistor or a quantum point contact coupled to a quantum
andthe qubit’s Hamiltoniando notcommute. The third
a bitisstudied. Wefindaunifieddescriptionofthestatisticsof
J time scale appears in the dynamics of the current in the
the monitored quantity, the current, in the regime of strong

8 measurement and expect this description toapply for a wide SET. Consider the probability distribution P(m,t) that
2 melectronshavetunneledthroughtheSETbytimet. It
class of quantum measurements. We derive the probability
distributions for the current and charge in different stages of wasshown[7]thatafter acertaintime tmeas information
]
l the process. In the parameter regime of the strong measure- about the state of the qubit can be extractedby reading
l
a ment the current develops a telegraph-noise behavior which outm. Asexpectedfromthebasicprinciplesofquantum
h can bedetected in thenoise spectrum. mechanicstheobservationofthequbitfirstofalldisturbs
- Introduction. The long-standing interest in funda- its state. Hence t τ . The measurementprocessis
s meas ϕ
≥
e mental questions of the quantum measurement received only effective if the mixing is slow, t t . In the
mix meas
m new impetus by the experimentalprogressin mesoscopic oppositelimitofstrongmixingtheinform≫ationaboutthe
. physics and increasing activities in quantum state engi- qubit’s state is lost before a read-out is achieved.
t
a neering. The basic idea is to use as meter a device, able The distribution function P(m,t) describes the statis-
m
to carry a macroscopic current, which is coupled to the tics of the charge which has tunneled. The distribution
- quantum system such that the conductance depends on ofthecurrentintheSETp(I,t)andcurrent-currentcor-
d
the quantum state. By monitoring the current one per- relations require, furthermore, the knowledge of corre-
n
o forms a quantum measurement, which, in turn, causes lations of the values of m at different times. In earlier
c a dephasing of the quantum system [1–3]. The dephas- papers on the statistics in a SET [7] or a QPC [8,9] the
[ ing has been demonstrated in an experiment of Buks et behavior of P(m,t) at times shorter than tmix was de-
1 al. [4] where a quantum dot is embedded in one arm of rived. Effects due to the additional knowledge acquired
v a ‘which-path’ interferometer. The current through a by an observer [10], and the possible influence of the
3 quantum point contact (QPC) in close proximity to the wave-functioncollapseonthe monitoredcurrent[8]were
2
dot suppresses the interference. However, since passing also discussed. Here we develop a systematic approach,
4
electronsinteractwiththecurrentonlyforashortdwell- based on the time evolution (Schr¨odinger equation) of
1
0 time in the dot, the meter fails to distinguish between thedensitymatrixofthe coupledsystem. Thisapproach
0 two possible paths of individual electron; only a reduc- allowsustostudyaveragesandcorrelatorsofthecurrent
0 tion of interference has been observed. This situation is and charge. Since, due to shot noise, instantaneous val-
t/ referred to as a weak measurement. uesofthecurrentfluctuatestrongly,westudythecurrent
a For a strong quantum measurement a different setting I¯, averaged over a finite time interval ∆t. We calculate
m
isneeded,whereaclosedquantumsystemisobservedby themixingtimeinaSETandderiveanalyticexpressions
d- ameter. Thenasufficientlylongobservationmayprovide forp(I¯,∆t,t),aswellasP(m,t),validonbothshortand
n information about the quantum state. This situation is long time scales. We study the noise spectrum of the
o realized when a single-electron transistor (SET) is cou- currentand find thatin the limit of strongmeasurement
c pledtoaJosephsonjunctionsingle-chargebox,whichfor (t <t )thelong-timedynamicsischaracterizedby
: meas mix
v suitableparametersservesasaquantumbit(qubit)[5,6]. telegraph noise,withjumpsbetweentwopossiblecurrent
Xi The analysis of the time evolution of the density matrix values, corresponding to two qubit’s eigenstates.
ofthecoupledsystemdemonstratesthemutualinfluence The results are of immediate experimental interest. A
r
a between qubit and meter, i.e. measurement and dephas- recent experiment demonstrated the quantum coherence
ing [7]. in a macroscopic superconducting electron box [11], but
The measurement process is characterized by three the coherence time was limited by the measuring device.
time scales. On the shortest, the dephasing time τϕ, The SET-based measurement should extend the coher-
thephasecoherencebetweentwoeigenstatesofthequbit ence time, which combined with experimental progress
is lost, while their occupation probabilities remain un- in fast measurement techniques [12] should increase the
changed. Latermeasurement-inducedtransitionsmixthe maximum number of coherent quantum manipulations.
eigenstates, changing their occupation probabilities on Master equation for the measurement by a SET. The
a time scale tmix > τϕ and erasing information about system of a qubit coupled to a SET is shown in Fig. 1.
1
ThequbitisaJosephsonjunctioninthe Coulombblock- current in the SET.
aderegime. Its dynamicsis limitedto atwo-dimensional At low temperatures and transport voltages only two
Hilbert space spanned by two charge states, with n = 0 charge states of the middle island of the SET, with N
or1extraCooperpaironasuperconductingisland. The and N +1 electrons,contribute to the dynamics. Trans-
island is coupled capacitively to the middle island of the lationalinvarianceinm-spacesuggeststheFouriertrans-
SET,influencing thetransportcurrent. TheSETis kept formation ρij(k) e−ikmρij(m). Expanding in the
N ≡ m N
in the off-state during manipulations on the qubit [7], tunneling term to lowest order, we obtain the following
P
with no dissipative current and no additional decoher- master equation (cf. Refs. [7,13]):
ence. To perform the measurement, the transport volt-
age is switched to a sufficiently high value, so that the ¯h d ρˆN + i[Hqb,ρˆN]
current starts to flow in the SET. dt(cid:18)ρˆN+1 (cid:19) i[Hqb+Hint,ρˆN+1]!
Γˇ eikΓˇ ρˆ
n (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) = −ΓˇL ΓˇR ρˆN . (2)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) N m(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:18) L − R (cid:19)(cid:18) N+1 (cid:19)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) I
V (cid:0)(cid:1) (cid:0)(cid:1)
x Here the operators
V V
g tr
Γˇ ρˆ Γ ρˆ+πα [ ,ρˆ] , (3)
FIG.1. The circuit of a qubit and a SET used as a meter. L ≡ L L Hint +
Γˇ ρˆ Γ ρˆ πα [ ,ρˆ] . (4)
R R R int +
≡ − H
The Hamiltonian of the system is given by [13]
represent the tunneling rates in the left and right junc-
H=HSET+HqNb+HT+Hψ . (1) tions,with αL/R ≡RK/(8π3RLT/R) being the tunnel con-
ductanceofthe junctionsinunitsofthe resistancequan-
The first term is the charging energy of the transis-
tum R = h/e2. The rates are fixed by the potentials
K
tor, quadratic in the charge Ne on the middle island,
µ , µ = µ +V of the leads: Γ = 2πα [µ (1
= E (N N )2. The induced charge N e is L R L tr L L L − −
HSET SET − 0 0 2N0)ESET] and ΓR =2παR[(1 2N0)ESET µR]. They
determined by the gate voltage V and other voltages − −
g define the tunneling rate Γ = Γ Γ /(Γ +Γ ) through
in the circuit. The Hamiltonian of the qubit, N = L R L R
Hqb the SET. The last terms in Eqs. (3,4) make these rates
+N , includesthe Hamiltonianoftheuncoupled
Hqb Hint sensitive to the qubit’s state.
qubit and the Coulomb interaction with the SET
Hqb The initial condition at the beginning of the measure-
N (splitintotwotermsforlaterconvenience.) Inthe
Hint ment, written down in the logical basis,
basisofthequbit’schargestatestheyaregivenby =
qb
H
e21n(eErqgbyσˆszc−aleEsJEσˆSxE)Ta,nEdqNb HanindtE=inEt ianrteNdσˆezt.erTmhineecdhabrygicnag- ρiNj(m,t=0)= |aa∗|b2 abb∗2 wN δm0 , (5)
pacitances in the circuit, while E is the Josephson cou- (cid:18) | | (cid:19)
J
pling. We considerthe eigenstatesof , 0 and 1 ,as describesthequbitinapurestatea0 +b1 andtheSET
qb
H | i | i | i | i
the logicalstates of the qubit. In this basis, is diag- in the zero-voltage equilibrium state. One can assume
qb
H
onal, with the level spacing ∆E (E2+E2 )1/2, while that w =1 and w =0.
≡ J qb N N+1
Ek E⊥ Reduction of the master equation. In general the dy-
Hint = Eii⊥nntt −Einikntt !, where Eiknt ≡ EintEqb/∆E and nmaamstiecrseoqfutahteioqnub(2it)’,sidsecnosmitpylimcaatterdixsi̺ˆn,cdeedscerpihbaesdinbgy(tdhee-
E⊥ E E /∆E. Theterm describestheFermions
int ≡ int J Hψ cay of the off-diagonal entries) and mixing (relaxation
in the island and electrodes of the SET, while gov-
T of the diagonal to their stationary values) may occur on
H
erns the tunneling in the SET. Here we assume weak
similar time scales, τ t . However, under suitable
ϕ mix
couplingtotheenvironment,withrelaxationslowerthan ≈
conditions the mixing is slow, which is the prerequisite
the SET-induced mixing. The opposite limit is dicussed
for a measurement process. This is the case, if the qubit
in Ref. [13].
operates in the regime with dominant charging energy:
ThefulldensitymatrixcanbereducedtoρiN m (t)by
jN′m′
tracing over microscopic degrees of freedom and keeping E , E +2E E . (6)
qb qb int J
| | | |≫
track only of N and m, the number of electrons which
have tunneled through the SET. Here i,j =0,1 refer to Weak (E ∆E) or strong (E ∆E) coupling to
int int
≪ ∼
a qubit’s basis. A closed set of equations can be derived the SET can be considered. In the latter case a faster
for ρij(m) ρiNm, the diagonal entries of the density measurement is achieved (see Eq. (11)).
N ≡ jNm
matrix in N and m [14]. Solving these equations, we We first analyze the dynamics without mixing, i.e. we
analyze the evolution of the reduced density matrix of put E⊥ = 0. In this case the time evolutions of ρij for
int N
the qubit, ̺ij(t) ρij(m,t), as well as P(m,t) the four different pairs of indices ij are decoupled from
≡ N,m N ≡
ρjj(m,t)andotherstatisticalcharacteristicsofthe eachother, each being characterizedby two eigenmodes.
N,j N P
P
2
The absence of mixing, further, implies the conservation equal-weightdistribution: ̺00(t) ̺11(t) exp( Γ t).
mix
− ∝ −
ofoccupationsofthelogicalstates̺ii,fori=0,1,andwe Thereforethe two-peak structure appears only in the in-
find two ‘Goldstone’ modes for k 1, with eigenvalues terval between t t Γ−1 . The measurement can
≪ be performed onmlyeaisf≤t ≤ mΓi−x1
1 meas ≪ mix
λii(k) iΓik fiΓik2 . (7) The measurement takes longer than the dephasing,
+ ≈ − 2
t τ . Such measurement can be called non-
meas ϕ
≫
Here Γi Γi Γi /(Γi + Γi ) are the tunneling rates efficient [10]: the information about the qubit is con-
≡ L R L R
throughtheSET,Γ0/1 =Γ πα Ek andΓ0/1 =Γ tained in the SET already after τϕ, but can be read out
πα Ek are the tunLnelingLra±tes inLthinetjunctioRns for Rtw∓o from the current only later.
R int The quantum measurement with a QPC can be de-
logicalstates(cf.Eqs.(3,4)),andfi 1 2Γi/(Γi +Γi )
≡ − L R scribed in a similar way. The Coulomb interaction of
are the Fano factors responsible for the shot noise re-
the qubit with the current results in two tunneling rates
duction. The other two eigenmodes decay fast, with the
Γ0/1 = Γ¯ δΓ/2 for two qubit’s states. Tracing out mi-
rates λii (Γi +Γi ). ±
− ≈− L R croscopic degrees of freedom one arrives at the master
The analysis of the eigenvalues, λ01(k) = [λ10( k)]∗,
± ± − equation[3] for the Fourier transformof the density ma-
ofthefouroff-diagonalmodesinij,revealsthedephasing
trix ρij(m), which can be rewritten as
time τ of the qubit by the measurement, i.e., the decay
ϕ
time of ̺01 = ρ01(k = 0)+ρ01 (k = 0). It is given
τby−1τ=ϕ−1w=Γ4Γ+EikwnNt2/(ΓLΓ+iΓnRt)h2eNiof+pE1pioknstit≪e caΓsLe.+ΓR, and ¯hddtρˆ+i[Hqb,ρˆ]=(cid:20)Γ¯ρˆ+ 21δΓ[σˆz,ρˆ]+(cid:21)(eik−1)
ϕ N L N+1 R (4τ )−1eik[σˆ ,[σˆ ,ρˆ]]. (12)
The picture is modified by the mixing at finite E⊥ . − ϕ z z
int
Wefindthatthemixingmaybetreatedperturbativelyif One can show that τ−1 1(√Γ0 √Γ1)2 is the decay
λ01 E⊥ , which turns to be the case if the condition ϕ ≡ 2 −
| ±|≫ int rateofρ01. Forρii theeigenvaluesaregivenby(7),with-
(6) holds. Then, in the secondorder,the degeneracybe-
out Fano factors. The measurement time (11) and the
tweenthelong-livingmodes(7)isliftedandthelong-time
dephasing time coincide, implying the 100% efficiency.
evolutionofthe occupationsρii(k)=ρii(k)+ρii (k) is
N N+1 Under the condition E max(∆E,τ−1) the per-
given by a reduced master equation, J ≪ ϕ
turbative treatment produces the reduced master equa-
d ρ00(k) ρ00(k) tion (8,9), with the mixing rate
=M , (8)
dt ρ11(k) red ρ11(k) τ
(cid:18) (cid:19) (cid:18) (cid:19) Γ =E2 ϕ . (13)
mix J1+∆E2τ2
λ00(k) 0 1 1 1 ϕ
Mred = +0 λ11(k) + 2Γmix −1 1 . (9)
(cid:18) + (cid:19) (cid:18) − (cid:19) A phenomenon, termed the Zeno (or watchdog) effect,
can be seen [3,8] in the limit τ−1 ∆E: the stronger
For the mixing rate, Γmix, we obtain ϕ ≫
is the measurement,quantified by τ−1, the weakeris the
ϕ
Γ rate of jumps between the eigenstates, Γ E2τ .
mix ≈ mix ≈ J ϕ
4ΓE2E2 The analysis of the SET mixing rate (10) in terms of
∆E2(∆E+2Ek )2+[ΓJ∆iEnt+Γ (∆E+2Ek )]2 . (10) the Zeno effect is more complicated. The rates τϕ−1 and
int R L int Γ depend in this case on several parameters and no
mix
To understand the role of the mixing we assume first simple relation between Γ and τ is found. However,
mix ϕ
Γ = 0 in Eqs. (8,9). Then, for the initial condition in the regime E Γ Γ ∆E , these two rates
mix int ≪ L ∼ R ≪ | |
(5)weobtainρ00(k)= a2eλ0+0(k)t, ρ11(k)= b2eλ1+1(k)t, change in opposite directions as functions of ΓL ΓR,
andP(k,t) P(m|,t|)e−ikm =ρ00(k)+ρ1|1|(k). From which is reminiscent of the Zeno physics. ∼
this we obt≡ain tmhe distribution P(m,t), which evolves Statistics of charge and current. The results of this
from a peak δP(m) at t = 0 into two peaks with weights section apply to the SET and QPC alike. The statisti-
a2 and b2, moving in m-space with velocities Γ0 and cal quantities studied depend on the initial density ma-
Γ| 1|, and |w|ith widths growing as 2fiΓit. The peaks trix (5): e.g., P(m,t) = P(m,t ρ0). In the two-mode
|
approximation (8,9) this reduces to a dependence on
separate after a time
p γ ̺00 ̺11 = a2 b2. We solve Eq. (8) to ob-
0
2f0Γ0+ 2f1Γ1 2 tain≡P(m,−t γ0)=T|r|qb−[U|(m|,t)ρ0], where U(m,t) is the
|
tmeas = p Γ0−Γp1 ! . (11) Γin0v/e1r=seΓ¯FourδiΓer/2traarnesfcolromse,otfhUe(rke,stu)lt≡inegxdpis[Mtrirbedu(tkio)nt]i.sIf
±
Thus measuring the charge m after t constitutes a
meas e−δm2/2fΓ¯t
strong quantum measurement [7]. However, at longer P(m,t γ )= P˜(m δm,t γ ) . (14)
times t > Γ−1 the mixing spoils this picture. In par- | 0 − | 0 2πfΓ¯t
mix δm
X
ticular, the occupations of the logical states relax to the
p
3
Thefirsttermintheconvolution(14)containstwodelta- asthesumoftheshot-andtelegraph-noisecontributions.
peaks, corresponding to two qubit’s logic states: At low frequencies ω Γ the latter becomes visible
mix
≪
ontopoftheshotnoiseasweapproachtheregimeofthe
P˜(m,t γ0)=Ppl m−Γ¯t , 1Γmixt γ0 strong measurement: Stelegraph/Sshot ≈4tmix/tmeas.
| δΓt/2 2
(cid:18) (cid:12) (cid:19)
+e−Γmixt/2 a2δ m Γ0t + b2δ(cid:12)m Γ1t . (15)
| | − | | (cid:12) − a P(m=t;t)(cid:1)t b p(I;(cid:1)t;t)
On the time s(cid:2)cale t(cid:0) Γ−(cid:1)1 the pe(cid:0)aks disap(cid:1)(cid:3)pear; in-
mix ≡ mix
stead a plateau arises. It is described by
Γ tmix tmix
Ppl(x,τ|γ0)=e−τ 2mδΓix I0 τ 1−x2 (cid:0)0 tmeas (cid:0)0
+(1+γ x)I τ 1n x(cid:16)2p/ 1 x(cid:17)2 , (16) (cid:0)1 tmeas<(cid:1)t<tmi(cid:0)x1
0 1
− −
c p(I;(cid:1)t;t) d p(I;(cid:1)t;t)
(cid:16) p (cid:17) p o
at x < 1 and P = 0 for x > 1. Here I , I are the
pl 0 1
| | | |
modified Bessel functions. At longer times the plateau
transforms into a narrow peak centered around m=Γ¯t.
tmix
The Gaussian in Eq. (14) arises due to shot noise. Its tmix
effect is to smear out the distribution (see Fig. 2a). (cid:0)0 (cid:0)0
We also calculate P2(m,t;m′,t′|ρ0), the joint proba- tmeas<(cid:1)t(cid:24)tm(cid:0)1ix (cid:1)t(cid:29)tmix (cid:0)1
bility to have m electrons at t and m′ electrons at t′.
This allows us to obtain the probability distribution FIG.2. Theprobabilitydistributionsofthecharge(a)and
current(b–d). P(m,t)in(a)isrescaled,sothatthepeaksdo
p(I¯,∆t,t ρ ) P (m+I¯∆t,t+∆t;m,t ρ ) (17) not move. Thetime-axis scale is logarithmic.
0 2 0
| ≡ |
m
X
To conclude, we have developed the master equation
of the current I¯ t+∆tI(t′)dt′ averaged over the time approach to study the statistics of currents in a SET or
≡ t
interval ∆t. The evolution is Markovian,and we obtain: aQPCasaquantummeter. Weevaluatetheprobability
R
P (m,t;m+∆m,t+∆t)=Tr [U(∆m,∆t)U(m,t)ρ ] distributions and the noise spectrum of the current.
2 qb 0
for ∆t >0. The derivation of p(I¯,∆t,t) thus reduces to We acknowledge discussions with Y. Blanter,
the calculation of the charge distribution (14) for differ- L. Dreher, S. Gurvitz, D. Ivanov, and A. Korotkov.
ent initial conditions:
p(I¯,∆t,t γ )=P(m=I¯∆t,∆t e−Γmixtγ ). (18)
| 0 | 0 [1] I.L. Aleiner, N.S. Wingreen, Y. Meir, Phys. Rev. Lett.
79, 3740 (1997).
The behavior of p(I¯,∆t,t) is shown in Fig. 2. A strong
[2] Y. Levinson, Europhys.Lett. 39, 299 (1997).
quantum measurement is achieved if tmeas < ∆t < tmix, [3] S.A. Gurvitz, Phys.Rev.B 56, 15215 (1997).
attimest<t (seeFig.2b). Inthiscasethemeasured
mix [4] E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and
currentis close to either Γ0 or Γ1, with probabilities a2 V. Umansky,Nature, 391, 871 (1998).
| |
and b2, respectively. At longer times a typical current [5] A.Shnirman,G.Sch¨on,andZ.Hermon,Phys.Rev.Lett.
patte|rn| is a telegraphsignaljumping betweenΓ0 andΓ1 79, 2371 (1997).
on a time scale t . If ∆t t the meter does not [6] Yu. Makhlin, G. Sch¨on, and A. Shnirman, Nature 386,
mix meas
≪
have enough time to extract the signal from the shot- 305 (1999).
noise background. At larger ∆t the meter-induced mix- [7] A. Shnirman and G. Sch¨on, Phys. Rev. B 57, 15400
(1998).
ing erases the information, partially (∆t t , Fig. 2c)
mix
∼ [8] S.A. Gurvitz, preprint,quant-ph/9808058.
or completely (∆t t , Fig. 2d), before it is readout.
≫ mix [9] L. Stodolsky, Phys.Lett. B 459, 193 (1999).
Thetelegraphnoisebehaviorisalsoseeninthecurrent
[10] A.N. Korotkov,Phys. Rev.B 60, 5737 (1999).
noise. Fourier transformation of the correlator [11] Y.Nakamura,Yu.A.Pashkin,andJ.S.Tsai,Nature398,
786 (1999).
hI(t)I(t′)iρ0 = mm′∂t∂t′P2(m,t;m′,t′|ρ0) (19) [12] R.J. Schoelkopf, P. Wahlgren, A. A. Kozhevnikov,
mX,m′ P. Delsing, and D. E. Prober, Science 280, 1238 (1998).
[13] Y. Makhlin, G. Sch¨on, and A. Shnirman, to be pub-
gives in the stationary case the noise spectrum,
lished in ”Exploring the Quantum-Classical Frontier.”
Eds. J.R.Friedman and S.Han, cond-mat/9811029.
e2δΓ2Γ
S (ω)=2e2fΓ¯+ mix , (20) [14] H. Schoeller and G. Sch¨on, Phys. Rev. B 50, 18436
I ω2+Γ2
mix (1994).
4