Master Equation without Markovian approximaton:
In the cases we are interested, the quantum systems are coupled to environments, such as bosonic fields or electron reservoirs. Although the environment is affected by the coupling, in general we can assume the coupling is weak, and the environment remains unchanged. This assumption will allow us to derive the master equation in a simple fashion.
Following the derivation of Y. Yamanoto and A. Imamoglu [10], we consider a system S interacting with a reservoir R via interaction V. If we assume initially (t=0) the system and reservoir is uncoupled, the initial density operator ρ(0) is then given by
)
Time evolution of ρ(t) in the interaction picture obeys the Liouville-von Neumann equation )]
If we further assume the number of degrees of freedom of the reservoir is very large, then the reduced density operator satisfies
)]
After integration and successive substitution one gets
)]])
Usually, it is impossible to solve the above equation exactly, and the Markov approximation is usually employed to solve it. In this project, however, we wish to go beyond this simple
approximation. Therefore, we just expand the commutators directly. In this case, one immediately finds that there are terms contain two-time correlation function, such as Vint(t)Vint(t')(t'). For the double QDs we are interested, the electron-phonon interaction is the main source of decoherence. The interaction term can actually be written as [11]
, electron-phonon coupling matrix element. To this point, one can apply the polaron transformation [12] to above equation, and it turns out that the appearance of the phase operators X, which are products of unitary displacement operator Dq Exp(zaq z*aq).
Therefore, the master equation will contain the terms like, . If the phonon environment is assumed in thermal equilibrium, the boson correlation is described by
, which depends on the time difference only. In this case, one may Laplace-transformed the whole equations into the Z-plane and probably solve them algebraically. To obtain the time-dependent dynamics of the qubit, it is usually a formidable task to perform the inverse Laplace transformation directly. Therefore, we suspect a numerical technique is required to resolve the dynamics at the last stage.
)
Non-Markovian coupling from the measurements:
As can be seen from Fig. 1, the left and right dots are coupled to the electron reservoirs.
To deal with the interactions, people usually assume that the chemical potential of the left (right) reservoir is shifted to positive (negative) infinity. In this case, the tunneling of the electron from the dot to the reservoir (or vice versa) can be approximated as an energy-independent tunneling rate, i.e. the Markovian limit. Unfortunately, it is usually not the case for real experiments.
Therefore, in this project, we will consider a more realistic situation: the tunneling density of state (DOS) from the dot to reservoir is energy-dependent. This means the tunneling of the electron is no more Markovian, and the corresponding equations of motions
have to be rewritten. In a previous work by Gurvitz and his co-worker [13], they considered the DOS with a Lorentzian form
4
where is the reservoir bandwidth. They introduced an auxiliary D-state that corresponds to a resonance state embedded in the reservoir to deal with such a non-Markovian problem.
Borrowing their idea, we will first consider the Lorentzian DOS to solve the dynamics of the qubit. The difference from Ref. [13] is that this Lorentzian DOS may also in return mix with the electron-phonon effect mentioned above. This will certainly increase the complexity of the problem and, of course, the difficulties of the numerical computations.
d
Counting Statistics for Quantum Shot Noise
Up to the present, we mentioned very few about the current-correlations measurements in this proposal. Actually, because electrons share the particle-wave duality with photons, one might expect fluctuations in the electrical current to play an important role. Current fluctuations due to the discreteness of the electrical charge are known as shot noise. Although the first observations of shot noise date from work on vacuum tubes in the 1920s, our quantum mechanical understanding of electronic shot noise has progressed more slowly than our understanding of photon noise. Much of the physical information shot noise contains has been appreciated only recently, from experiments on nanoscale conductors [14], where classical mechanics breaks down. At that scale, shot noise can reveal a rich variety of details about charge transport.
In a quantum conductor in nonequilibrium, electronic current noise originates from the dynamical fluctuations of the current being away from its average. To study correlations between carriers, we will relate the electron dynamics with the reservoir operators by introducing the degree of freedom n as the number of electrons that have tunneled through the barrier [15] and define the expectation values, O(n)
i0,L,RTrbath n,iOˆ(t)n,i with
n
O n
O ( ) . This leads to a system of equations of motion,
)
L/R eans the occupation probability of the 0
R or by time t,
n 0 L R )
(t), the shot noise spectrum can be determined from the Mac-Donald formula [16]
P=∣L>< ∣. Together with the total probability of finding n electrons in the collect P (t)=n (n)(t)+ n (n)(t)+ n (n
efore, a careful and self-consistent treatment of this part will probably be the main difficulty.
oto and A. Imamoglu, Mesoscopic quantum optics (Wiley-Interscience, New
72, 233301 (2005).
where I represents the current.
Notes that in the above calculations, the assumption for the electron tunneling is Markovian.
Since we are interested in the non-Markovian regime, the derivations might be invalid.
Fortunately, in a previous work by Gurvitz and his co-worker [13], they introduced an additional
“auxiliary state” to represents the non-Markovian effect, and the whole formulation for the counting statistics still holds. We will thus adopt their idea and apply it to our two-dot or four-dot device. However, one should also keep in mind that in our case there is another non-Markovian source form the dissipative (phonon) environment. Ther
Reference
[10] Y. Yamam York, 1999).
[11] T. Brandes, Phys. Rep. 408, 315 (2005).
[12] G. D. Mahan, Many-Particle Physics, (Plenum, New York, 1990).
[13] B. Elattari and S. A. Gurvitz, Phys. Rev. Lett. 84, 2047 (2000).
[14] Y. M. Blanter and M. Buttiker, Phys. Rep. 336, 1 (2000).
[15] Y. N. Chen, D. S. Chuu, and S. J. Cheng, Phys. Rev. B
onald, Rep. Prog. Phys. 12, 56 (1948).
[16] D. K. C. MacD