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Conductance dependent suppression of current partition noise in mesoscopic electron branching circuits
ELSEVIER
Physica B 210 (1995) 37-42
Conductance dependent suppression of current partition noise in mesoscopic electron branching circuits R.C. Liu*, Y. Y a m a m o t o 1 E. L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA
Received 19 November 1993
Abstract
Inelastic scattering subject to local charge conservation realizes the transition from ballistic to dissipative electron division, and suppresses the current partition noise in mesoscopic electron branching circuits. The relevant parameter for noise suppression is the output conductance and not the location of the scatterers. We find the noise in an electron beam splitter (Y-branch) decreases linearly (nonlinearly) with decreasing conductance. Physically, the suppression mechanism is a natural implementation of the measurement and feedback control of electron numbers.
The current partition noise in a macroscopic branching circuit is determined by Kirchhoff's current law. If a (noiseless) current I from a highimpedance current source is divided into two branches having resistances R1 and R2, the spectral densities of the divided currents and their correlation are
Sv~(~o) + Sv2(~o) SI1(°))
=
SI2(°))
=
-
SllI2((°) =
(Rl + R2) 2
'
(1) where Sw(og) = 4Ri[he~/2 + hco/(exp(hog/kBT) 1)] is the voltage noise spectral density associated with the resistance R~. The first term in the bracket is a quantum mechanical zero-point fluctuation and the second term is a Johnson-Nyquist *Corresponding author. 1Also at: NTT Basic Research Laboratories,3-9-11 Midori-cho, Musashino-shi, Tokyo 180,Japan.
thermal noise. The current partition noise can be made smaller than the shot noise value, Sshot(og) = 2eli, for the low-frequency limit (o9 ~ kBT/h) and high resistance limit (Ri >>2kBT/eI). This suppressed current partition noise was the key ingredient for the generation by a semiconductor laser of light with greatly reduced intensity fluctuations [1]. As indicated in Eq. (1), the current partition noise in the two branches are negatively correlated, a fact confirmed by the observation of a (quantum) negative intensity correlation from two parallel coupled light emitting diodes
E2]. The current partition noise is completely suppressed in the limit of zero temperature and zero frequency (DC) for a finite resistance. This conclusion drawn from Kirchhoff's law breaks down in a mesoscopic branching circuit when ballistic electron division occurs. A single electron wave is split into two branches with finite electron number partition noise, (A(Noutl-Nout2)2), and phase
0921-4526/95/$09.50 © 1995 ElsevierScienceB.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 0 2 9 6 - 7
38
R.C Liu, E Yamamoto/Physica B 210 (1995) 3~42
partition noise, (A(q~o,t~ - ~bo,t~)2) (Nr and q~ represent, the electron number and phase in output branch 7). These together satisfy Heisenberg's minimum uncertainty product, (A(Nout~--Nout2)2) (A(~bo.t~-q go,u) 2) = 1 which is derived from a spin state analogy [3]. Unlike the zero-point fluctuations of Eq. (1), the current partition noise in mesoscopic branching circuits does not disappear in the DC limit. It originates from the complementarity relation and the fact that the divided electron waves in the two branches are phase coherent. Previous work on two-terminal mesoscopic conductors of the (elastic scattering dominant) diffusive regime and (inelastic scattering dominant) dissipative regime has indicated that inelastic scattering is essential for reaching the macroscopic regime of suppressed transport noise [4, 5]. In this paper, we demonstrate that the transition to recover Kirchhoff's division law is realized by introducing inelastic scattering subject to local charge conservation. As long as the scattering is completely inelastic, its location and distribution after the division are immaterial; the output conductance associated with the inelastic scattering can parameterize the noise suppression completely. Our results apply to a beam splitter and a Ybranch division at zero temperature and frequency. For simplicity, we assume the branching circuit consists of identical, one-dimensional, singlechannel conductors connected to ideal electron reservoirs. The scattering approach to mesoscopic transport [6, 7] allows us to express the fluctuating current in any reservoir ), in the Langevin form:
eE
I~ = ~ (1
-
T~)Izr --
T~apa
l
+ 8I~,
(2)
scattering matrix for transport between reservoirs fl and ~, T~a = Tr(o~psra) is the transmission probability between reservoirs fl and 7, and B is the noise bandwidth. Current noise is characterized by the Fano factor, F = (AN2)/I ( N ) I , where N = I z / e is the total number of electrons measured in a reservoir during a certain time interval, z. For a ballistic two-terminal conductor at zero temperature, F = 0 since every emitted electron travels unimpeded to its receiving reservoir. When a 50-50 beam splitter or an impedancematched Y-branch (Fig. 1) is used to divide a noiseless flux of electrons in a ballistic regime, the Fano factor becomes F = ½(half-Poissonian). This can be understood as an interference between the incident electron state and the vacuum fluctuations from the open port (beam splitter) or receiving reservoirs (Y-branch). The in-phase component of the vacuum fluctuation induces the phase partition noise and the quadrature component induces the number (current) partition noise [3]. Note that this view-
(a)
,, L _ _ /
"xJ
(b) ~A
I Partition ! I jlI 1/2 current in each branch
(c)
I
where 2e2 B
t, oo
( 8 I ~ I ~ ) = --h--- ~ L
Refet'ence
x Tr [(8~r5~ - o~o~p) x (8~8~, - ~)~o~a)].
(3)
fr(E) is the Fermi Dirac distribution, /t r is the
chemical potential, 8It is the current noise when all chemical potentials are held constant, o~ is the
- ILl. /2(
1)
I I ! !
Energy
dE f~(E)(1--f~(E))
ou,2
Emitting Reservoir
] gB Inelastic Scattering Reservoirs
Receiving Reservoirs
Fig. 1. A 50-50 beam splitter geometry (a) and an impedance matched symmetricY-branch (b) with M( = 2) inelastic scattering reservoirsin each branch. Solid arrows indicate the incident electron states, and the hatched arrows indicate the vacuum fluctuations. (c) A schematic of the zero temperature energy distributions for electronsemitted fromeach of the reservoirsfor the geometriesabove.
R.C Liu, Y. Yamamoto/Physica B 210 (1995) 37-42
point suggests a correlated electron wavepacket instead of a "squeezed vacuum" can be used to control the partition noise [8]. For either ballistic geometry, the output conductance is GQ/2, where GQ is the quantum unit of conductance, e2/h (ignoring spin degeneracy). Take first the case of partition by a beam splitter. Inelastic scattering subject to local charge conservation is added according to the approach of Beenakker and Biittiker [5]. The dual channel 4 × 4 scattering matrix [9] used to incorporate these inelastic scattering reservoirs (infinite impedance voltage probes) introduces no elastic scattering: all incident electrons lose excess energies. Because of the complete thermalization, all electrons finally absorbed in the receiving reservoirs originate from the closest scattering reservoir. Since the transmission probability between the receiving reservoir and the closest scattering reservoir is one, the output conductance is just determined by their chemical potential difference. For one scattering reservoir in a branch, the output conductance is reduced to GQ/4, which includes contributions from the current division (a factor of 1) and the energy dissipation at the scattering reservoir (additional factor of ½). For M scattering reservoirs in a branch, the average adjacent chemical potential difference is further decreased, and the output conductance is reduced to Gout = GQ/2(M + 1). Local charge conservation implies that the fluctuations in the number of electrons absorbed by a scattering reservoir ~ are completely compensated for by the fluctuations in the number of reemitted electrons. This Condition forces the scattering reservoir's chemical potential to fluctuate with magnitude 6/A, determined by solving the set of linear equations {AL = 0: 0t = scat1 . . . . , scatu I, where I~ is given by Eq. (2). Consequently, 6p~ is a function of the intrinsic current fluctuations in the M scattering reservoirs (8Iscat. . . . . ,8Iscat,~) which exist when local charge conservation is not enforced. The resulting current noise in a receiving reservoir y is scats4
AI r = ~ I ~ - (e/h)
~
T~8/~(8Iscat . . . . . . ~IsCatM),
39
The physical mechanism responsible for the suppression of the quantum current partition noise is similar to the quantum nondemolition (QND) measurement and feedback control that suppresses photon flux noise [10]. At the scattering reservoir, the incident electrons lose their excess energy to phonons and produce the fluctuating chemical potential in the voltage probe. The voltage probe "measures" the fluctuating incident electron number. This can be considered as a QND measurement of the electron number, since the local charge conservation requirement preserves the electron number in each branch while the electron phase is completely randomized (back action noise). Due to the loss of the electron's kinetic energy, this is not an ideal QND measurement; rather it falls in the same class of QND as ideal photon counting followed by ideal reemission of photons. If the chemical potential in the voltage probe increases due to the initial excess current, the subsequent current from the emitting reservoir to the scattering reservoir decreases, and thus suppresses the fluctuating chemical potential. Also, if the chemical potential initially decreases, the subsequent current increases. This can be considered as a feedback control of a subsequent electron flow based on the QND measurement outcome. In our model for the inelastic scattering reservoir, the reemitted electrons equally flow backward toward the emitting reservoir (counteracting the fluctuating incident current), and forward toward the receiving reservoir (carrying the residual current noise according to the scattering reservoir's chemical potential fluctuation). The residual noise can be reduced further by adding more inelastic scattering reservoirs. The electron number fluctuation from the last scattering reservoir is calculated by Eqs. (2)-(4) as (2Gout/Go)~)Npart , where ~Npart = (Npart/2) 1/2 --[Goz(iz 1 -1~2)/4e] 1/2 (half-Poissonian) is the deviation from the average electron number Npart imposed by the partition for the case of M = 0 (ballistic division). Since the average received electron number depends linearly on the output conductance,
at =scat i
(4) where SI r is the intrinsic current noise in the receiving reservoir.
(2G,,uq2/~ N ~2 G-O-O' ~ part, F :
Gout(,t/1 _ t / 2 ) T / e -
Gout GQ -- 2 ( M
1 +
1~)"
(5)
40
R.C. Liu, E Yamamoto/Physica B 210 (1995) 3 ~ 4 2
The Fano factor for the beam splitter calculated by Eq. (5) are plotted by filled squares in Fig. 2 for the case of zero, one, two, four, ten and twenty scattering reservoirs in each branch. We note that no importance is given to the distance between scattering reservoirs; only their net effect on the conductance plays a role in the suppression. Even though this result appears model specific, the trend toward greater noise suppression with smaller conductance (higher resistance) should be a general property of dissipative division in real mesoscopic systems. Also shown in Fig. 2 is the detrimental effect of elastic scattering on the current partition noise. The noise is also calculated from scattering theory, Eq. (3), mentioned earlier. The increase of resistance by elastic scatterers is not associated with energy dissipation and so there is no measurement-feedback control involved. In this sense, it is similar to a lossy optical transmission line which attenuates excess photon number noise and also destroys squeezed
. . . .
i
. . . .
i
. . . .
I
. . . .
I
. . . .
" •0.75
cq
L.
~
Ballistic I3q'v'ffmn Limit
0.5
o
~
, 0.25
• •
0 0
Inelastic Scattering
m.O. • , . . . . . . . . . . . . . . . . . . . 0.1 0.2 0.3
0.4
0.5
Output C o n d u c t a n c e (units o f e2/h) Fig. 2. The Fano factor for one output branch of the beam splitter versus the normalized output conductance, Gout/G o. Ballistic transport, corresponding to Gout = GQ/2, exhibits halfPoissonian noise (F = ½). The square points are calculated by the scattering theory (3) and (4) for one, two, four, 10, and 20 inelastic scattering reservoirs. The solid line, calculated by the scattering theory (3), indicates the effect of increasing elastic scattering.
photon number noise, making the final statistics approach the Poissonian limit [11]. The three-port scattering matrix used to model the Y-branch partition [12] allows an incident electron's wavefunction to be split into two branches without reflection. It also permits the two output branches to couple with each other. As a result, in addition to the feedback regulation described above, there is also a push-pull mechanism operating to equalize the chemical potential difference between the scattering reservoirs in the two output branches. If we assume an equal number of scattering reservoirs in both branches, then symmetry implies that no current will flow between them on average. However, when there are excess electrons in one branch, the chemical potential of the scattering reservoir in that branch increases, while it decreases in the other. This potential difference induces a net current flow between the two branches. Consequently, excess electrons in the former branch are scattered backward into the latter branch, reducing the partition noise. The calculations for a symmetric Y-branch case confirm a stronger (nonlinear) suppression with decreasing output conductance as compared to the beam splitter case. The results are plotted in Fig. 3 for zero, one, and two scattering reservoirs in each arm. If the two branches are asymmetric in the number of scattering reservoirs, the output conductances of each differ. In the limiting case of no scatterers (ballistic transport) in one branch and one scatterer in the other, the partition noise is still suppressed in the ballistic branch due to the interbranch compensating current. In this event, the output conductance of a single branch is not suitable for determining the degree of partition noise suppression. The general trend of increased inelastic scattering and decreased partition noise is well parameterized by the overall output conductance of the parallel circuit, as also shown in Fig. 3. For the ballistic beam splitter, symmetric or asymmetric Y-branch geometries, the two output branches are perfectly anticorrelated, i.e. ANout, = - ANout2 [7]. As inelastic scattering increases in the beam splitter and symmetric Ybranch, a perfect anticorrelation remains because the total incident current is kept constant (in the
R.C. Liu, Y. Yamamoto/Physica B 210 (1995) 37 42 0.5
. . . , . . , , , . . , . , • • •
A
Z
~v
0.4
¢-qA V
, . r . "
Symmetric Y-Branch Asymmetric Y-B Output 1 Asymmetric Y-B Output 2
0.3
O
o
0.2
~. 0.1 © i
0
,
. 0.2 . .
. 0.4 . .
06
08
Parallel Output Conductance (units of e2/h) Fig. 3. The Fano factor for one receiving reservoir of the Ybranch circuit versus normalized parallel output conductance. The square points correspond to the symmetric Y-branch, and the diamond and triangular points correspond to the asymmetric Y-branch case when output 1 has one more scattering reservoir than output 2.
limit of zero temperature and zero frequency). For the asymmetric Y-branch however, the incident current fluctuates and the perfect anticorrelation is lost. For the particular example of two scattering reservoirs in one branch, and one scattering reservoir in the other, the usual correlation parameter has a value of - 1.4. Finally, it must be emphasized that the physical mechanism responsible for the partition noise suppression by inelastic scattering reservoirs concerns only the resistance associated with energy dissipation. It does not matter where the scattering reservoirs are placed after the division. This fact may find an interesting application in the design of future mesoscopic devices and circuits with adjacent ballistic and dissipative parts. Furthermore, the requirement of Gou t "~ Go for suppressing the current partition noise may be connected to the source impedance requirements for Coulomb blockade [13]. In order for Coulomb blockade ira a single tunnel junction to work, con-
41
tinuous charging by the lead is indispensible. This requires that the source and lead conductance Gs is much smaller than Go . Both macroscopic squeezing (squeezed light generation) rl, 2] and microscopic single electron-photon conversion (heralded single photon generation) [14] are based on the Coulomb blockade and the macroscopic bias circuit [15]. The "macroscopic bias circuit" does not necessarily mean a physically macroscopic size; rather even a single-channel mesoscopic electron waveguide with intentionally introduced inelastic scattering would operate as a "macroscopic circuit." In contrast to an "artificial" configuration of QND measurement and feedback in quantum optics [10], the inelastic scattering and local charge conservation are a natural realization of a measurement-feedback mechanism for partition noise suppression. Though our calculations are model specific, the trend is quite general, as we have shown for both independent (beam splitter) and mutually coupled (Y-branch) output leads. R.C. Liu would like to thank the National Defence Science and Engineering Graduate program and the Office of Naval Research for supporting his research.
References [1] W.H. Richardson, S. Machida and Y. Yamamoto, Phys. Rev. Lett. 66 (1991) 2867; Y. Y a m a m o t o and H. A. Haus, Phys. Rev. A 45 (1992) 6596. [2] P.J. Edwards and G.H. Pollard, Phys. Rev. Lett. 69 (1992) 1757; E. Goobar, A. Karlsson, G. Bj6rk and P.J. Rigole, Phys. Rev. Lett. 70 (1993) 437. [3] R.C. Liu and Y. Yamamoto, Phys. Rev. B 49 (1994) 10520. [4] A. Shimizu and M. Ueda, Phys. Rev. Lett. 69 (1992) 1403. I-5] C.W.J. Beenakker and M. Biittiker, Phys. Rev. B 46 (1992) 1889. [6] G B . Lesovik, J E T P Lett. 49 (1989) 592; B. Yurke and G.P. Kochanski, Phys. Rev. B 41 (1990) 8184; M. Biittiker, Phys. Rev. Lett. 65 (1990) 2901; C.W.,I. Reenakker and H. van Houten, Phys. Rev. B 43 (1991) 12066. [7] Th. Martin and R. Landauer, Phys. Rev. B 45 (1992) 1742. [8] Masahiro Kitagawa and Masahito Ueda, Phys. Rev. Lett. 67 (1991) 1852. [9] M. Biiniker, Phys. Rev. B 33 (1986) 3020. [10] Y. Yamamoto, N. Imoto and S. Machida, Phys. Rev. A 33 (1986) 3243; H.A. Haus and Y. Yamamoto, Phys. Rev. A 34 (1986) 270.
42
R.C. Liu, Y. Yamamoto/Physica B 210 (1995) 37-42
[11] Y. Yamamoto, S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa and G. Bj6rk,° in: Progress in Optics, ed. E. Wolf (Elsevier, Amsterdam, 1990) p. 89. [12] M. Biittiker, Phys. Rev. B 32 (1985) 1846. [13] D.V. Averin and K.K. Likharev, in: Mesoscopic Phenomena in Solids, eds. B.L. Altshuler, P.A. Lee and R.A.
Webb (Elsevier, Amsterdam, 1991). [14] A. Imamoglu, Y. Yamamoto and P. Solomon, Phys. Rev. B 46 (1992) 9555; A. Imamoglu and Y. Yamamoto, Phys. Rev. B 46 (1992) 15982. [15] A. Imamoglu and Y. Yamamoto, Phys. Rev. Lett. 70 (1993) 3327.
Physica B 210 (1995) 37-42
Conductance dependent suppression of current partition noise in mesoscopic electron branching circuits R.C. Liu*, Y. Y a m a m o t o 1 E. L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA
Received 19 November 1993
Abstract
Inelastic scattering subject to local charge conservation realizes the transition from ballistic to dissipative electron division, and suppresses the current partition noise in mesoscopic electron branching circuits. The relevant parameter for noise suppression is the output conductance and not the location of the scatterers. We find the noise in an electron beam splitter (Y-branch) decreases linearly (nonlinearly) with decreasing conductance. Physically, the suppression mechanism is a natural implementation of the measurement and feedback control of electron numbers.
The current partition noise in a macroscopic branching circuit is determined by Kirchhoff's current law. If a (noiseless) current I from a highimpedance current source is divided into two branches having resistances R1 and R2, the spectral densities of the divided currents and their correlation are
Sv~(~o) + Sv2(~o) SI1(°))
=
SI2(°))
=
-
SllI2((°) =
(Rl + R2) 2
'
(1) where Sw(og) = 4Ri[he~/2 + hco/(exp(hog/kBT) 1)] is the voltage noise spectral density associated with the resistance R~. The first term in the bracket is a quantum mechanical zero-point fluctuation and the second term is a Johnson-Nyquist *Corresponding author. 1Also at: NTT Basic Research Laboratories,3-9-11 Midori-cho, Musashino-shi, Tokyo 180,Japan.
thermal noise. The current partition noise can be made smaller than the shot noise value, Sshot(og) = 2eli, for the low-frequency limit (o9 ~ kBT/h) and high resistance limit (Ri >>2kBT/eI). This suppressed current partition noise was the key ingredient for the generation by a semiconductor laser of light with greatly reduced intensity fluctuations [1]. As indicated in Eq. (1), the current partition noise in the two branches are negatively correlated, a fact confirmed by the observation of a (quantum) negative intensity correlation from two parallel coupled light emitting diodes
E2]. The current partition noise is completely suppressed in the limit of zero temperature and zero frequency (DC) for a finite resistance. This conclusion drawn from Kirchhoff's law breaks down in a mesoscopic branching circuit when ballistic electron division occurs. A single electron wave is split into two branches with finite electron number partition noise, (A(Noutl-Nout2)2), and phase
0921-4526/95/$09.50 © 1995 ElsevierScienceB.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 0 2 9 6 - 7
38
R.C Liu, E Yamamoto/Physica B 210 (1995) 3~42
partition noise, (A(q~o,t~ - ~bo,t~)2) (Nr and q~ represent, the electron number and phase in output branch 7). These together satisfy Heisenberg's minimum uncertainty product, (A(Nout~--Nout2)2) (A(~bo.t~-q go,u) 2) = 1 which is derived from a spin state analogy [3]. Unlike the zero-point fluctuations of Eq. (1), the current partition noise in mesoscopic branching circuits does not disappear in the DC limit. It originates from the complementarity relation and the fact that the divided electron waves in the two branches are phase coherent. Previous work on two-terminal mesoscopic conductors of the (elastic scattering dominant) diffusive regime and (inelastic scattering dominant) dissipative regime has indicated that inelastic scattering is essential for reaching the macroscopic regime of suppressed transport noise [4, 5]. In this paper, we demonstrate that the transition to recover Kirchhoff's division law is realized by introducing inelastic scattering subject to local charge conservation. As long as the scattering is completely inelastic, its location and distribution after the division are immaterial; the output conductance associated with the inelastic scattering can parameterize the noise suppression completely. Our results apply to a beam splitter and a Ybranch division at zero temperature and frequency. For simplicity, we assume the branching circuit consists of identical, one-dimensional, singlechannel conductors connected to ideal electron reservoirs. The scattering approach to mesoscopic transport [6, 7] allows us to express the fluctuating current in any reservoir ), in the Langevin form:
eE
I~ = ~ (1
-
T~)Izr --
T~apa
l
+ 8I~,
(2)
scattering matrix for transport between reservoirs fl and ~, T~a = Tr(o~psra) is the transmission probability between reservoirs fl and 7, and B is the noise bandwidth. Current noise is characterized by the Fano factor, F = (AN2)/I ( N ) I , where N = I z / e is the total number of electrons measured in a reservoir during a certain time interval, z. For a ballistic two-terminal conductor at zero temperature, F = 0 since every emitted electron travels unimpeded to its receiving reservoir. When a 50-50 beam splitter or an impedancematched Y-branch (Fig. 1) is used to divide a noiseless flux of electrons in a ballistic regime, the Fano factor becomes F = ½(half-Poissonian). This can be understood as an interference between the incident electron state and the vacuum fluctuations from the open port (beam splitter) or receiving reservoirs (Y-branch). The in-phase component of the vacuum fluctuation induces the phase partition noise and the quadrature component induces the number (current) partition noise [3]. Note that this view-
(a)
,, L _ _ /
"xJ
(b) ~A
I Partition ! I jlI 1/2 current in each branch
(c)
I
where 2e2 B
t, oo
( 8 I ~ I ~ ) = --h--- ~ L
Refet'ence
x Tr [(8~r5~ - o~o~p) x (8~8~, - ~)~o~a)].
(3)
fr(E) is the Fermi Dirac distribution, /t r is the
chemical potential, 8It is the current noise when all chemical potentials are held constant, o~ is the
- ILl. /2(
1)
I I ! !
Energy
dE f~(E)(1--f~(E))
ou,2
Emitting Reservoir
] gB Inelastic Scattering Reservoirs
Receiving Reservoirs
Fig. 1. A 50-50 beam splitter geometry (a) and an impedance matched symmetricY-branch (b) with M( = 2) inelastic scattering reservoirsin each branch. Solid arrows indicate the incident electron states, and the hatched arrows indicate the vacuum fluctuations. (c) A schematic of the zero temperature energy distributions for electronsemitted fromeach of the reservoirsfor the geometriesabove.
R.C Liu, Y. Yamamoto/Physica B 210 (1995) 37-42
point suggests a correlated electron wavepacket instead of a "squeezed vacuum" can be used to control the partition noise [8]. For either ballistic geometry, the output conductance is GQ/2, where GQ is the quantum unit of conductance, e2/h (ignoring spin degeneracy). Take first the case of partition by a beam splitter. Inelastic scattering subject to local charge conservation is added according to the approach of Beenakker and Biittiker [5]. The dual channel 4 × 4 scattering matrix [9] used to incorporate these inelastic scattering reservoirs (infinite impedance voltage probes) introduces no elastic scattering: all incident electrons lose excess energies. Because of the complete thermalization, all electrons finally absorbed in the receiving reservoirs originate from the closest scattering reservoir. Since the transmission probability between the receiving reservoir and the closest scattering reservoir is one, the output conductance is just determined by their chemical potential difference. For one scattering reservoir in a branch, the output conductance is reduced to GQ/4, which includes contributions from the current division (a factor of 1) and the energy dissipation at the scattering reservoir (additional factor of ½). For M scattering reservoirs in a branch, the average adjacent chemical potential difference is further decreased, and the output conductance is reduced to Gout = GQ/2(M + 1). Local charge conservation implies that the fluctuations in the number of electrons absorbed by a scattering reservoir ~ are completely compensated for by the fluctuations in the number of reemitted electrons. This Condition forces the scattering reservoir's chemical potential to fluctuate with magnitude 6/A, determined by solving the set of linear equations {AL = 0: 0t = scat1 . . . . , scatu I, where I~ is given by Eq. (2). Consequently, 6p~ is a function of the intrinsic current fluctuations in the M scattering reservoirs (8Iscat. . . . . ,8Iscat,~) which exist when local charge conservation is not enforced. The resulting current noise in a receiving reservoir y is scats4
AI r = ~ I ~ - (e/h)
~
T~8/~(8Iscat . . . . . . ~IsCatM),
39
The physical mechanism responsible for the suppression of the quantum current partition noise is similar to the quantum nondemolition (QND) measurement and feedback control that suppresses photon flux noise [10]. At the scattering reservoir, the incident electrons lose their excess energy to phonons and produce the fluctuating chemical potential in the voltage probe. The voltage probe "measures" the fluctuating incident electron number. This can be considered as a QND measurement of the electron number, since the local charge conservation requirement preserves the electron number in each branch while the electron phase is completely randomized (back action noise). Due to the loss of the electron's kinetic energy, this is not an ideal QND measurement; rather it falls in the same class of QND as ideal photon counting followed by ideal reemission of photons. If the chemical potential in the voltage probe increases due to the initial excess current, the subsequent current from the emitting reservoir to the scattering reservoir decreases, and thus suppresses the fluctuating chemical potential. Also, if the chemical potential initially decreases, the subsequent current increases. This can be considered as a feedback control of a subsequent electron flow based on the QND measurement outcome. In our model for the inelastic scattering reservoir, the reemitted electrons equally flow backward toward the emitting reservoir (counteracting the fluctuating incident current), and forward toward the receiving reservoir (carrying the residual current noise according to the scattering reservoir's chemical potential fluctuation). The residual noise can be reduced further by adding more inelastic scattering reservoirs. The electron number fluctuation from the last scattering reservoir is calculated by Eqs. (2)-(4) as (2Gout/Go)~)Npart , where ~Npart = (Npart/2) 1/2 --[Goz(iz 1 -1~2)/4e] 1/2 (half-Poissonian) is the deviation from the average electron number Npart imposed by the partition for the case of M = 0 (ballistic division). Since the average received electron number depends linearly on the output conductance,
at =scat i
(4) where SI r is the intrinsic current noise in the receiving reservoir.
(2G,,uq2/~ N ~2 G-O-O' ~ part, F :
Gout(,t/1 _ t / 2 ) T / e -
Gout GQ -- 2 ( M
1 +
1~)"
(5)
40
R.C. Liu, E Yamamoto/Physica B 210 (1995) 3 ~ 4 2
The Fano factor for the beam splitter calculated by Eq. (5) are plotted by filled squares in Fig. 2 for the case of zero, one, two, four, ten and twenty scattering reservoirs in each branch. We note that no importance is given to the distance between scattering reservoirs; only their net effect on the conductance plays a role in the suppression. Even though this result appears model specific, the trend toward greater noise suppression with smaller conductance (higher resistance) should be a general property of dissipative division in real mesoscopic systems. Also shown in Fig. 2 is the detrimental effect of elastic scattering on the current partition noise. The noise is also calculated from scattering theory, Eq. (3), mentioned earlier. The increase of resistance by elastic scatterers is not associated with energy dissipation and so there is no measurement-feedback control involved. In this sense, it is similar to a lossy optical transmission line which attenuates excess photon number noise and also destroys squeezed
. . . .
i
. . . .
i
. . . .
I
. . . .
I
. . . .
" •0.75
cq
L.
~
Ballistic I3q'v'ffmn Limit
0.5
o
~
, 0.25
• •
0 0
Inelastic Scattering
m.O. • , . . . . . . . . . . . . . . . . . . . 0.1 0.2 0.3
0.4
0.5
Output C o n d u c t a n c e (units o f e2/h) Fig. 2. The Fano factor for one output branch of the beam splitter versus the normalized output conductance, Gout/G o. Ballistic transport, corresponding to Gout = GQ/2, exhibits halfPoissonian noise (F = ½). The square points are calculated by the scattering theory (3) and (4) for one, two, four, 10, and 20 inelastic scattering reservoirs. The solid line, calculated by the scattering theory (3), indicates the effect of increasing elastic scattering.
photon number noise, making the final statistics approach the Poissonian limit [11]. The three-port scattering matrix used to model the Y-branch partition [12] allows an incident electron's wavefunction to be split into two branches without reflection. It also permits the two output branches to couple with each other. As a result, in addition to the feedback regulation described above, there is also a push-pull mechanism operating to equalize the chemical potential difference between the scattering reservoirs in the two output branches. If we assume an equal number of scattering reservoirs in both branches, then symmetry implies that no current will flow between them on average. However, when there are excess electrons in one branch, the chemical potential of the scattering reservoir in that branch increases, while it decreases in the other. This potential difference induces a net current flow between the two branches. Consequently, excess electrons in the former branch are scattered backward into the latter branch, reducing the partition noise. The calculations for a symmetric Y-branch case confirm a stronger (nonlinear) suppression with decreasing output conductance as compared to the beam splitter case. The results are plotted in Fig. 3 for zero, one, and two scattering reservoirs in each arm. If the two branches are asymmetric in the number of scattering reservoirs, the output conductances of each differ. In the limiting case of no scatterers (ballistic transport) in one branch and one scatterer in the other, the partition noise is still suppressed in the ballistic branch due to the interbranch compensating current. In this event, the output conductance of a single branch is not suitable for determining the degree of partition noise suppression. The general trend of increased inelastic scattering and decreased partition noise is well parameterized by the overall output conductance of the parallel circuit, as also shown in Fig. 3. For the ballistic beam splitter, symmetric or asymmetric Y-branch geometries, the two output branches are perfectly anticorrelated, i.e. ANout, = - ANout2 [7]. As inelastic scattering increases in the beam splitter and symmetric Ybranch, a perfect anticorrelation remains because the total incident current is kept constant (in the
R.C. Liu, Y. Yamamoto/Physica B 210 (1995) 37 42 0.5
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Parallel Output Conductance (units of e2/h) Fig. 3. The Fano factor for one receiving reservoir of the Ybranch circuit versus normalized parallel output conductance. The square points correspond to the symmetric Y-branch, and the diamond and triangular points correspond to the asymmetric Y-branch case when output 1 has one more scattering reservoir than output 2.
limit of zero temperature and zero frequency). For the asymmetric Y-branch however, the incident current fluctuates and the perfect anticorrelation is lost. For the particular example of two scattering reservoirs in one branch, and one scattering reservoir in the other, the usual correlation parameter has a value of - 1.4. Finally, it must be emphasized that the physical mechanism responsible for the partition noise suppression by inelastic scattering reservoirs concerns only the resistance associated with energy dissipation. It does not matter where the scattering reservoirs are placed after the division. This fact may find an interesting application in the design of future mesoscopic devices and circuits with adjacent ballistic and dissipative parts. Furthermore, the requirement of Gou t "~ Go for suppressing the current partition noise may be connected to the source impedance requirements for Coulomb blockade [13]. In order for Coulomb blockade ira a single tunnel junction to work, con-
41
tinuous charging by the lead is indispensible. This requires that the source and lead conductance Gs is much smaller than Go . Both macroscopic squeezing (squeezed light generation) rl, 2] and microscopic single electron-photon conversion (heralded single photon generation) [14] are based on the Coulomb blockade and the macroscopic bias circuit [15]. The "macroscopic bias circuit" does not necessarily mean a physically macroscopic size; rather even a single-channel mesoscopic electron waveguide with intentionally introduced inelastic scattering would operate as a "macroscopic circuit." In contrast to an "artificial" configuration of QND measurement and feedback in quantum optics [10], the inelastic scattering and local charge conservation are a natural realization of a measurement-feedback mechanism for partition noise suppression. Though our calculations are model specific, the trend is quite general, as we have shown for both independent (beam splitter) and mutually coupled (Y-branch) output leads. R.C. Liu would like to thank the National Defence Science and Engineering Graduate program and the Office of Naval Research for supporting his research.
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R.C. Liu, Y. Yamamoto/Physica B 210 (1995) 37-42
[11] Y. Yamamoto, S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa and G. Bj6rk,° in: Progress in Optics, ed. E. Wolf (Elsevier, Amsterdam, 1990) p. 89. [12] M. Biittiker, Phys. Rev. B 32 (1985) 1846. [13] D.V. Averin and K.K. Likharev, in: Mesoscopic Phenomena in Solids, eds. B.L. Altshuler, P.A. Lee and R.A.
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