Proposal for detection of non-Markovian decay via current noise
Yueh-Nan Chen1,*
and Guang-Yin Chen21Department of Physics and National Center for Theoretical Sciences, National Cheng-Kung University, Tainan 701, Taiwan 2Institute of Physics, National Chiao-Tung University, Hsinchu 300, Taiwan
共Received 2 August 2007; published 14 January 2008兲
We propose to detect non-Markovian decay of an exciton qubit coupled to multimode bosonic reservoir via shot-noise measurements. Nonequilibrium current noise is calculated for a quantum dot embedded inside a
p-i-n junction. An additional term from non-Markovian effect is obtained in the derivation of noise spectrum.
As examples, two practical photonic reservoirs, photon vacuum with the inclusion of cut-off frequency and surface plasmons, are given to show that the noise may become super-Poissonian due to this non-Markovian effect. Utilizing the property of super-radiance is further suggested to enhance the noise value.
DOI:10.1103/PhysRevB.77.035312 PACS number共s兲: 73.20.Mf, 03.65.Yz, 72.70.⫹m, 73.63.⫺b
I. INTRODUCTION
Due to rapid progress of quantum information science, great attention has been focused on the dynamics of systems interacting with their surroundings. Radiative decay of a two-level atom may be one of the most obvious examples in this issue and can be traced back to such early works as that of Einstein in 1917.1 While Markovian approximation is widely adopted to treat decoherence and relaxation prob-lems, non-Markovian dynamics of qubit systems have at-tracted increasing attention lately.2
Turning to solid state systems, an exciton in a quantum dot 共QD兲 can be viewed as a two-level system. Radiative properties of QD excitons, such as super-radiance3 and
Pur-cell effect,4have attracted great attention during the past two
decades. Utilizing QD excitons for quantum gate operations has also been demonstrated experimentally.5 With the
ad-vances of fabrication technologies, it is now possible to em-bed QDs inside a p-i-n structure,6such that the electron and
hole can be injected separately from opposite sides. This allows one to examine the exciton dynamics via electrical currents.7
Recently, the interest in measurements of shot noise in quantum transport has risen owing to the possibility of ex-tracting valuable information not available in conventional dc transport experiments.8In this work, we propose to detect
non-Markovian decay of an exciton qubit via the current noise of a QD p-i-n junction. Without making Markovian approximation to the exciton-boson interaction, we analyti-cally show that the Fano factor共zero-frequency noise兲 may become super-Poissonian. As examples, two practical photo-nic environments, photon vacuum with the inclusion of cut-off frequency and surface plasmons, are given to show this non-Markovian effect. To enhance the noise value, we fur-ther suggest utilizing the property of super-radiance.
II. MODEL
QDs can now be embedded in a p-i-n junction, such that many applications can be accomplished by electrical control. As shown in Fig. 1, we wish to see non-Markovian effect between the system and reservoir via measurements of elec-trical currents. For simplicity, both the hole and electron
res-ervoirs of the p-i-n junction are assumed to be in thermal equilibrium. For the physical phenomena we are interested in, the Fermi level of the p共n兲-side hole 共electron兲 is slightly lower 共higher兲 than the hole 共electron兲 subband in the dot. After a hole is injected into the hole subband in the QD, the
n-side electron can tunnel into the exciton level because of
the Coulomb interaction between the electron and hole. Thus, we may introduce the three dot states:兩0典=兩0,h典, 兩↑典 =兩e,h典, and 兩↓典=兩0,0典, where 兩0,h典 means there is one hole in the QD,兩e,h典 is the exciton state, and 兩0,0典 represents the ground state with no hole and electron in the QD. One might argue that one cannot neglect the state兩e,0典 for real devices since the tunable variable is the applied voltage. This can be resolved by fabricating a thicker barrier on the electron side so that there is little chance for an electron to tunnel in advance.7Thus, the couplings of the dot states to the electron
and hole reservoirs are to be described by the standard tunnel Hamiltonian HT=
兺
q 共Vqcq †兩0典具↑兩 + W qdq †兩0典具↓兩 + H.c.兲, 共1兲 where cq and dq are the electron operators in the right and left reservoirs, respectively. Vqand Wqcouple the channels q of the electron and the hole reservoirs. The interaction be-tween the exciton qubit and its bosonic environment is writ-ten asFIG. 1. 共Color online兲 Schematic view of a QD p-i-n junction with its exciton coupled to a bosonic environment.
Hex-bosonic=
兺
kDkbk
†兩↓典具↑兩 + H.c. = 兩↓典具↑兩X + 兩↑典具↓兩X†, 共2兲 where X =兺kDkbk†, bk† denotes the creation operator of the
bosonic reservoir, and Dkdescribes the system-reservoir
cou-pling.
With Eqs.共1兲 and 共2兲, one can now write down the
equa-tion of moequa-tion for the reduced density operator
d
dt共t兲 = − Trres
冕
0t
dt
⬘
兵HT共t兲 + Hex-bosonic共t兲,关HT共t⬘
兲+ Hex-bosonic共t
⬘
兲,⌶˜ 共t⬘
兲兴其, 共3兲 where⌶˜ 共t⬘
兲 is the total density operator. Note that the trace in Eq.共3兲 is taken with respect to both bosonic and electronicreservoirs. If the couplings to the electron and hole reservoirs are weak, it is reasonable to assume that the standard Born-Markov approximation with respect to the electronic cou-plings is valid. In this case, multiplying Eq. 共3兲 by nˆ↑
=兩↑典具↑兩 and nˆ =兩↓典具↓兩, the equations of motions can be writ-↓ ten as t
冉
具nˆ 典↑ t 具nˆ 典↓ t冊
=冕
dt⬘
冉
− A共t − t⬘
兲 具nˆ 典↑ t⬘ A共t − t⬘
兲 具nˆ 典↓ t⬘冊
+冋
−⌫L −⌫L 0 −⌫R册
冉
具nˆ 典↑ t 具nˆ 典↓ t冊
+冉
⌫L 0冊
, 共4兲 where⌫L= 2兺qVq 2␦共 ↑−q↑兲, ⌫R= 2兺qWq 2␦共 ↓−q↓兲, and =ប0=↑−↓ is the energy gap of the QD exciton. Here,A共t−t
⬘
兲⬅C共t−t⬘
兲+C*共t−t⬘
兲 can be viewed as the 共bosonic兲reservoir correlation function with the function C defined as
C共t−t
⬘
兲⬅具XtXt⬘†典0
. The appearance of the two-time correla-tion is attributed to that in the derivacorrela-tion of Eq.共4兲; we only
assume the Born approximation to the bosonic reservoir, i.e., the Markovian one is not made.
One can now define the Laplace transformation for real z,
C共z兲 ⬅
冕
0 ⬁ dte−zteitC共t兲, n↑共z兲 ⬅冕
0 ⬁ dte−zt具nˆ 典↑ t, etc., z⬎ 0, 共5兲and transform the whole equations of motion into z space,
n↑共z兲 = − A共z兲n↑共z兲/z +⌫L
z 关1/z − n↑共z兲 − n↓共z兲兴, n↓共z兲 = A共z兲n↓共z兲/z −⌫R
z n↓共z兲. 共6兲
These equations can then be solved algebraically, and the tunnel current from the hole-side barrier, Iˆ =−e⌫RR 具nˆ 典↓ t, can
in principle be obtained by performing the inverse Laplace transformation. Depending on the complexity of the correla-tion funccorrela-tion C共t−t
⬘
兲 in the time domain, this can be a for-midable task which can, however, be avoided if one directly seeks the quantum noise.In a quantum conductor in nonequilibrium, electronic cur-rent noise originates from the dynamical fluctuations of the current around its average. To study correlations between carriers, we relate the exciton dynamics with the hole reser-voir operators by introducing the degree of freedom n as the number of holes that have tunneled through the hole-side barrier, and write
n˙0共n兲共t兲 = − ⌫Ln0共n兲共t兲 + ⌫Rn↓共n−1兲共t兲,
n˙↑共n兲共t兲 + n˙↓共n兲共t兲 = 共⌫L−⌫R兲n0共n兲共t兲. 共7兲 Equation 共7兲 allows us to calculate the particle current and
the noise spectrum from Pn共t兲=n0 共n兲共t兲+n
↑
共n兲共t兲+n
↓
共n兲共t兲, which gives the total probability of finding n electrons in the col-lector by time t. In particular, the noise spectrum SIR can be
calculated via the MacDonald formula,9,10
SIR共兲 = 2e2
冕
0 ⬁ dt sin共t兲d dt关具n 2共t兲典 − 共t具I典兲2兴, 共8兲 where dtd具n2共t兲典=兺nn2P·n共t兲. With the help of countingstatistics,10one can obtain
SI
R共兲 = 2eI
再
1 +⌫R冋
A共i兲⌫L
− A共i兲⌫L⌫R+共A共i兲 + i兲共⌫L+ i兲共⌫R+ i兲
+ A共− i兲⌫L
− A共− i兲⌫L⌫R+共A共− i兲 − i兲共⌫L− i兲共⌫R− i兲
册
冎
, 共9兲
where A共z兲⬅C共z兲+C*共z兲.
As can be seen from Eq.共9兲, the noise spectrum indeed
contains the information of memory effect, i.e., A共i兲 and
A共−i兲. However, it is not easy to see how it affects the
noise. We thus take the zero-frequency limit共→0兲, and an analytical solution with physical meaning is obtained:
F = SI R共= 0兲/2e具I典 = 1 −2⌫L⌫R兵Re关A共0兲兴⌫L+ Re关A共0兲兴共Re关A共0兲兴 + ⌫R兲其 兵Re关A共0兲兴⌫R+⌫L共Re关A共0兲兴 + ⌫R兲其2 + 2 Im
冋
冏
A共iw兲 w冏
w=0册
⌫L 2⌫ R 2 兵Re关A共0兲兴⌫R+⌫L共Re关A共0兲兴 + ⌫R兲其2 . 共10兲If one makes Markovian approximation to the bosonic reser-voir, the third term in Eq.共10兲 vanishes and the Fano factor
共F兲 is further reduced to usual sub-Poissonian result. The question here is whether this additional term is positive or not, such that the noise feature may become super-Poissonian. To answer this, let us now consider real bosonic environments.
III. SURFACE PLASMONS
The collective motions of an electron gas in a metal or semiconductor are known as the plasma oscillations. The nonvanishing divergence of the electric field Eជ,ⵜ·Eជ⫽0, in the bulk material gives rise to the well known bulk plasma modes, characterized by the plasma frequency p
=共4n0e2/m兲1/2, where m and e are the electronic mass and charge and n0is the electron density. In the presence of sur-face, however, the situation becomes more complicated. Not only the bulk modes are modified, but also the surface modes can be created.11Like the bulk modes, surface plasmons can
be excited by incident electrons or photons.12 Many works
were devoted to the study of radiative decay into surface plasmons.13 Recently, it is now possible to fabricate QDs
evanescently coupled to surface plasmons, such that en-hanced fluorescence is observed.14 Based on these
develop-ments, it is plausible to assume that the QD p-i-n junction is close to a metal surface. This allows one to examine non-Markovian effect from surface plasmons.
When a semiconductor QD is near a metal surface, the vector potential to the QD exciton can be decomposed into contributions from s- and p-polarized photons and surface plasmons as follows:15
A共rជ,t兲 = As共rជ,t兲 + Ap共rជ,t兲 + Asp共rជ,t兲. 共11兲
Figure2shows the corresponding radiative decay rates of a QD exciton in front of a silver surface. It is evident that at short distances radiative decay is dominated by surface mons. Since we are interested in the effect from surface plas-mons, we thus keep the QD in this regime, and consider only the interaction from surface plasmons:
Hex-sp=
兺
k冉
4k 2 បAcpk冊
1/2冋
兩↑典具↓兩冉
kˆ · pˆ + ik 0· pˆ冊
akeik·−0 z + H.c.册
. 共12兲Here, we have chosen cylindrical coordinates rជ=共ជ, z兲 in the half-space z艌0; kជ is a two-dimensional wave vector in the metal surface of area A. ak is the annihilation operator of
surface plasmon and pˆ is the transition dipole moment. The surface-plasmon frequencykand the parameters,0and pk,
are given by k 2 =1 2p 2 + ck2−
冉
1 4p 4 + c2k4冊
1/2 , 0= k2−k 2/c2, pk= ⑀ 4共 k兲 − 1 关−⑀共0兲 − 1兴1/2 1 ⑀2共 k兲 , 共13兲 where ⑀共k兲=1−p 2/ k2 is the dielectric function of the metal. By replacing Hex-bosonic with Hex-sp, one can go through the procedure to obtain the current noise.
The shot-noise spectrum of InAs QD excitons is numeri-cally displayed in Fig.3, where the tunneling rates,⌫Land
⌫R, are assumed to be equal to 10−40 and 10−30,
respec-tively. The plasma oscillation energyបpof silver and
exci-0 1 2 3 4 5 d@λê2πD 1 2 3 4 5 6 γ
FIG. 2. 共Color online兲 Radiative decay rate of QD exciton in front of a silver surface with distance d共in units of /2, where is the wavelength of the emitted photon兲. The plasma oscillation energyបpof silver and exciton band gap energyប0are 3.76 and
1.39 eV共Ref.6兲, respectively. The black dashed 共solid兲 line repre-sents the decay into the surface plasmons共photons兲 as the exciton dipole moment pˆ is oriented perpendicular to the surface. The red lines are the case for pˆ parallel to the surface.
-0.001 0 0.001 ω -40 -200 20 40 -0.0005 0 0.0005 ω 0.99 1 1.01 1.02 1.03 1.04 SRI @ω Dê 2Ie
FIG. 3. 共Color online兲 Shot-noise spectra of QD excitons in front of a silver surface. The black, red, and blue lines represent the results of various dot-surface distances: d = 0.1, 0.045, and 0.03共in units of/2⬇1423 Å兲, respectively. The inset shows the corre-sponding curves of the imaginary part of A共i兲.
ton band gap energyប0 are 3.76 and 1.39 eV. One knows from Fig.2that there is no essential difference in physics for different orientations of the exciton dipole moment. There-fore, in plotting the figure the dipole moment pˆ is assumed to be along zˆ direction for simplicity. Without making Markov-ian approximation, the black, red, and blue lines represent the results for different dot-surface distances: d = 0.1, 0.045, and 0.03共in unit of /2⬇1423 Å兲, respectively. As seen, the Fano factor gradually changes from sub-Poissonian noise to super-Poissonian one as the QD is moving toward the surface. This proves that the additional term in Eq.共10兲 can
change the noise feature. The inset of Fig. 3 numerically shows the imaginary part of A共i兲. As the QD is closer to the silver surface, the slope becomes steeper, which coincides with the analytical result of Eq.共10兲.
The reasons for super-Poissonian noise actually depend on the details of the device structures, for example, positive correlations due to resonant tunneling states,16 noise
en-hancement due to quantum entanglement,17 spin-flip
cotun-neling processes,18non-Markovian coupling between dot and
leads,19 and quantum shuttle effect.20The underlying
physi-cal picture in our case may be similar to a recent work by Djuric et al.21 They considered the tunneling problem
through a QD connected coherently to a nearby single-level dot, which is not connected to the left and right leads. In this case, the coherent hopping to the nearby dot also gives an extra “positive” term to the Fano factor. The explanation is that the coming electron can either tunnel out of the original dot directly, or travel to the nearby dot and come back again. This indirect path is the origin of the super-Poissonian noise. In our case, as the exciton decays into surface plasmon, the non-Markovian effect from the plasmon reservoir may reex-cite it now and then, such that the Fano factor is enhanced. One notes that this kind of enhancement due to quantum coherence has recently been observed in the tunneling through a stack of coupled quantum dots22 and explained
theoretically.23
IV. CUTOFF FREQUENCY
To see whether this non-Markovian effect is general, let us return to the old quantum electrodynamic共QED兲 problem: spontaneous emission. Under Markovian approximation, the emission rate of a two-level atom in free space can be easily obtained via Fermi’s golden rule and is given by ␥ = 2兺q兩Dq兩2␦共0− c兩q兩兲, where Dqis the atom-reservoir cou-pling strength. Its frequency counterpart is written as ⌬ =P兰dq兩Dq兩2/共0− c兩q兩兲, where P denotes the principal inte-gral. To remove the divergent problem from the integration, one can, for example, include the concept of cutoff fre-quency to renormalize the frefre-quency shift.24In this case, the
exciton-photon coupling is described by the Hamiltonian
Hex-ph=
兺
k 1 关1 + 共k/B兲2兴2 Dkbk †兩↓典具↑兩 + H.c., 共14兲 where the introduced Lorentzian cutoff contains the nonrel-ativistic cutoff frequencyB⬇c/aB, with aBbeing theeffec-tive Bohr radius of the exciton.25 Replacing H
ex-bosonic by
Hex-ph, one can obtain the corresponding noise spectrum straightforwardly. As shown in Fig. 4, the Fano factor is sub-Poissonian共black line兲 under Markovian approximation, while it may become super-Poissonian 共as shown by the dashed and red lines兲 with the consideration of non-Markovian effect from the Lorentzian cutoff.
One also finds that the magnitude of the Fano factor de-pends on the cutoff frequencyB. With the increasing ofB,
the Fano factor becomes larger共the dashed line兲. This phe-nomenon allows one to examine the cutoff frequency in QED. However, one might argue that the value of the super-Poissonian noise is extremely small and may not be observ-able in real experiments. To overcome this obstacle, we sug-gest making use of the property of collective decay 共super-radiance兲.26For example, one can, instead of the QD,
insert a quantum well 共QW兲 into the p-i-n junction. The interaction between the共two-dimensional兲 QW exciton and the photon can be written as27
H
⬘
=兺
qnm兺
kz 1 关1 + 共k/B兲2兴2 e m0c冑
2បc 共q2+ k z 2兲1/2v ⫻共⑀qkz·nm兲bqkzcqnm † + H.c., 共15兲 where nm=冑
N兺
Fnm* 共兲冕
d2wc*共−兲共− iប ⵜ 兲wv共兲. 共16兲 Here, cq†and bqkzstand for the exciton and photon operators. ⑀qkz is the polarization vector of the photon. Fnm共兲 is thetwo-dimensional hydrogenic wave function of the exciton
-0.1 -0.05 0.05 0.1 ω 1 1.0005 SRI @ω Dê 2Ie -0.1 -0.05 0.05 0.1 ω 1 1.05 1.1
FIG. 4. 共Color online兲 Shot-noise spectra of QD excitons in the presence of Lorentzian cutoff. Sub-Poissonian noise represented by the black line is the result of Markovian approximation. Super-Poissonian noise共red and dashed lines兲 is the consequence of non-Markovian effect. To plot the figure, the exciton spontaneous life-time共=1/␥兲 used here is 1.3 ns, and the cutoff frequency for red 共dashed兲 line is 9⫻1016共1.2⫻1017兲 Hz. Inset: noise increased by
the enhancement of the effective dipole moment共400 times兲 via super-radiance.
with quantum number n and m. wc共兲 and wv共兲 are,
respec-tively, the Wannier functions for the conduction band and the valence band.
With the Hamiltonian in Eq.共15兲, the radiative decay rate
of the QW exciton can be obtained straightforwardly
␥qnm⬃␥0共/d兲2, 共17兲
where␥0is the decay rate of a lone exciton, is the wave-length of the emitted photon, and d is the lattice constant of the material. The enhanced rate in Eq.共17兲 implies the
co-herent contributions from the lattice atoms within half a wavelength or so. In other words one can say that the effec-tive dipole moment of the QW exciton is enhanced by a factor of共/d兲2.27From Eq.共10兲, Fig.2, and inset of Fig.3,
we know that an enhanced rate somehow implies a larger Fano factor. Consider the real experimental values,28the
ob-served enhancement is around several hundred times the lone exciton. We thus plot the Fano factor in the inset of Fig.4. As can be seen, the value of super-Poissonian noise is greatly enhanced by super-radiance. This gives a better chance to observe the mentioned effect. Another possible candidate for the enhancement is the uniform QD arrays.29Within the
col-lective decay area defined by2, the effective dipole moment
may also be enhanced by a factor of共/r兲2, where r is the dot-lattice constant.
Finally, we note that recent advances in fabrication nano-technologies have made it possible to grow high quality nanowires,30 in which cavity QED phenomena can be
re-vealed via surface plasmons.31It is likely that similar effects
will appear if the QD p-i-n junction is coupled to the channel
plasmons. Even more, since the dispersion relation in
cylin-drical interface is much more complex共for example, it con-tains both real and virtual modes兲,32 the corresponding
shot-noise spectra are expected to give more information about the non-Markovian effect. Further investigations in this di-rection certainly put such a system more useful in the fields of quantum transport and cavity QED.
ACKNOWLEDGMENTS
We would like to thank T. Brandes and E. Schöll at Tech-nische Universität Berlin, J. Y. Hsu, and S. H. Chang at Institute Electro-Optical Engineering for helpful discussions. This work was supported partially by the National Science Council of Taiwan under Grant No. NSC 95-2112-M-006-031-MY3.
*yuehnan@mail.ncku.edu.tw
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