2.3 Dynamical Coulomb blockade
2.3.1 Ohmic environment
dω ω
ReZ(ω) RK
eiωt−1
1−e−β~ω, (2.77) whereZ(ω) is the total impedance of the circuit as seen from the junction,
Z(ω) = 1
iωCT +Zex−1(ω). (2.78) Finally, the current through the junction is given by
IL→R(V) =e[ΓL→R(V)−ΓR→L(V)]. (2.79) Because of ΓR→L(V) = ΓL→R(−V), we nd
IL→R(V) = 1 eRT
Z
dE[f(E−eV)−f(E+eV)]P(E). (2.80) P(E) can be interpreted as the probability to emit the energy E into the envi-ronment during a tunneling process. It is normalized and obeys the detailed balance relation
P(−E) = e−βEP(E), (2.81)
which means that the emission of energy is exponentially surpressed with respect to absorption. For a low external impedance, Zex → 0 we have P(E) = δ(E) and the I-V characteristic becomes linear,
IL→R(V) = 1
eRT [f(−eV)−f(eV)] = V
RT. (2.82)
Similarly, for a high impedance, ReZ(ω) RK, P(E) becomes Gaussian centered aroundEC with a width proportional to the temperature. At low temperatures this results in a suppression of the current for|eV|< EC, i.e. a Coulomb gap.
2.3.1 Ohmic environment
In Ch. 5 we consider an ohmic environment which consists of a resistance R in parallel with a capacitanceC. The total impedance then reads
Z(ω) = 1
iωCΣ+ 1/R, (2.83)
0.0 0.5 1.0 1.5 2.0 2.5
Figure 2.5: a) Energy transfer probability P(E) for an ohmic environment with resistance R. b) Conductance spectrum for a tunnel junction with an ohmic impedance. As R increases, the zero-bias feature develops from a dip into a gap.
RK =h/e2 ≈26kΩand EC is the charging energy. Gaussian centered around the charging energyEC =e2/(2CΣ).
The eect of the impedance can best be seen in the dierential conductance, dI
dV =− 1 RT
Z
dE[f0(E+eV) +f0(E−eV)]P(E). (2.85) In the zero temperature limit we have
f0(E) = −Θ(−E), (2.86)
andP(E) = 0vanishes forE <0because of Eq. (2.81). The dierential conductance then becomes
The last equation follows from the fact thatP(E) is normalized. For small positive energies,P(E)follows a power law with exponentρ−1, whereρ= 2R/RK [73]. For small voltages, the dierential conductance is then proportional to|V|ρ.
In Fig. 2.5b we plotdI/dV atkT = 0.01EC as a function ofV for dierent values ofR. Because of the nite temperature, the conductance at small voltages does not go all the way to zero. For R RK a dip-like zero-bias anomaly is visible in the spectrum. AsR becomes larger than RK, it develops into a gap of the order of EC. In Ch. 5 we use the DCB eect to explain the conductance spectra obtained by scanning tunneling spectroscopy measurements on small Pb islands grown on dierent substrates. As it turns out, the islandsubstrate interface can be modeled as an ohmic environment. The resistance R and capacitance C of the interface are used as tting parameters to explain the data.
Chapter 3
Entanglement entropy in dynamic quantum-coherent conductors
The concept of entanglement entropy is currently at the forefront of condensed matter physics [61, 62, 63]. Originally developed in the context of black hole physics [104], entanglement entropy was later adopted in the quantum information sciences to quantify the degree of entanglement between two parties [105]. In recent years, it has also been recognized as a useful quantity in condensed matter systems, for instance to investigate quantum critical systems [106, 107, 108, 109, 110, 111], quantum quenches [112, 113, 114, 115, 116, 117], topologically ordered states [118, 119, 120], and strongly correlated systems [121]. One important nding is that in gapless one-dimensional fermionic systems, the entanglement entropy depends logarithmically on the system size [122], while in quenched systems the role of spatial extent is played by time [109, 116].
Despite the theoretical interest, the measurement of entanglement entropy re-mains challenging, since its denition does not refer directly to any physical observ-able. Here we focus on a proposal that relates the entanglement entropy between two reservoirs of non-interacting electrons to the FCS of transferred charge [69, 70, 71, 72].
In this approach, the entanglement and the Rényi entropies are expressed as series in the cumulants of the FCS. Since charge uctuations in nano-scale electronics are now being detected experimentally [36, 44, 45, 46, 47, 48, 49, 50, 51, 123, 124, 125, 126, 127], these relations provide a means to measure the entanglement entropy in quantum-coherent conductors.
We consider the schematic setup in Fig. 3.1a, showing two Fermi seas connected by a constriction whose transmission can be controlled in a time-dependent manner.
To be specic, such a setup can be realized experimentally by connecting two elec-tronic leads via a QPC, as illustrated in Fig. 3.1b. We concentrate on time-dependent situations [128, 129], where either the transmission of the QPC is modulated in time, or designed pulses are applied to the leads to generate clean single-particle excitations (levitons) on top of the Fermi sea, following the proposal by Levitov and co-workers [26, 27, 28] and recent experiments reported in Refs. [29, 30].
D=0.5 D(t) QPC
a) b)
1
Figure 3.1: Entanglement entropy in quantum-coherent conductors. a) Two Fermi seas connected via a barrier with a time-dependent transmission D(t). The reser-voirs exchange particles, leading to many-body entanglement of the Fermi seas. b) The barrier may consist of a QPC connecting two nano-scale electrodes. Here, a clean single-particle excitation (a leviton) is partitioned on the QPC tuned to half transmission.
We are interested in a setup where the QPC is operated as a quantum switch that is opened and closed in a periodic manner. In this case, we show that a logarith-mic growth of the entanglement entropy as predicted by conformal eld theory [122, 106, 109] should be observable under suitable experimental conditions. Im-portantly, we nd that the logarithmic growth can be inferred from a measurement of only the rst few cumulants of the FCS. We also consider the entanglement entropy produced by partitioning levitons on the QPC, as shown in Fig. 3.1b, and by inter-fering two levitons on the QPC tuned to half transmission. In this case, we identify a Hong-Ou-Mandel-like suppression of the entanglement entropy as a function of the dierence in arrival times at the QPC [130, 25].
Problems concerning nite-time FCS for time-dependent systems are dicult to treat analytically. Here we employ a numerically exact scheme based on the tight-binding model shown in Fig. 2.1. The FCS has previously been investigated for such systems without an external driving [83, 84, 85, 131]. We extend the approach to time-dependent Hamiltonians (see also [132, 133, 134]). With this method, we can investigate the inuence of external modulations on the cumulants and the entan-glement entropy as functions of time, and we can identify the number of cumulants needed to reliably approximate the entanglement entropy. We focus here on meso-scopic conductors, but our tight-binding model may also describe cold atoms in optical lattices [59, 135, 136, 137, 138].
This chapter is structured as follows. In Sec. 3.1 we introduce the entanglement and the Rényi entropies and reiterate how for non-interacting fermions they can be expressed as series in the cumulants of the FCS. We illustrate these ideas with a simple example involving only two fermions. We then show how the entropies can be evaluated for non-interacting fermions in the framework of our tight-binding model introduced in Sec. 2.1. In Sec. 3.2 we consider the quantum switch. We investigate the increase of the entanglement entropy upon opening the QPC and how it is aected by a nite opening time, nite temperatures, nite bias, and a
non-perfect transmission. In Sec. 3.3 we consider levitons generated by applying designed pulses to the leads. In this case, we identify a Hong-Ou-Mandel-like suppression of the entanglement entropy by interfering two levitons on the QPC. Finally, in Sec. 3.3 we summarize our results. Technical details are provided in App. A.
3.1 Entanglement entropy and full counting statis-tics
The entanglement entropy of a quantum many-body system is dened with re-spect to a partitioning of the system into a subsystem and its complement. Here we analyze the entanglement entropy between particles in two electronic leads connected by a QPC as illustrated in Fig. 3.1. To dene the entanglement entropy of the right lead (R), we introduce the reduced density matrix
ˆ
ρR=TrL[ˆρ], (3.1)
obtained by tracing out the degrees of freedom of the left lead (L), where the density matrix of the full system is denoted asρ. An analog denition holds for the left lead.ˆ The entanglement entropy of the right lead is dened as the von Neumann entropy of ρˆR,
SR=−TrR[ˆρRlog ˆρR], (3.2) and equivalently for the left lead. If the full system is in a pure stateρˆ=|ΨihΨ|, it holds thatSR =SL.
A larger class of entanglement measures is provided by the Rényi entropies SR(ν) = 1
1−ν log (TrR[ˆρνR]) (3.3) of orderν [139]. We focus here on integer ordersν = 2,3, . . ., while the entanglement entropy is obtained in the limitν →1. In the following we treat identical leads such that SL(ν) =SR(ν). We therefore skip the subscript and always evaluate the entropies in the right lead. For a system with N particles in a pure state, the entanglement entropy takes on values between zero for a product state andNlog 2 for a maximally entangled state [105].
The denitions of the entanglement and Rényi entropies Eqs. (3.2) and (3.3) are general and can be applied to a variety of quantum systems. However, they do not refer to any physical observables, making their measurement a dicult task.
Here we consider non-interaction electrons. The entanglement and Rényi entropies can then be expressed in terms of the cumulants of the FCS of transferred charge (see Sec. 2.1.2). Specically, the entanglement entropy can be obtained as the limit [70, 71]
S = lim
K→∞SK (3.4)
of the series
SK =
K+1X
m=1
am(K)Cm. (3.5)
The cut-o dependent coecients are am(K) =
2PK k=m−1
S1(k,m−1)
k!k m even,
0 m odd , (3.6)
whereS1(n, m)are the unsigned Stirling numbers of the rst kind and the cumulants are given by Eq. (2.13). The series (3.5) provides an increasingly accurate lower bound to the exact entanglement entropy and it converges from below to the exact value as more cumulants are included [70, 71].
The Rényi entropies can also be related to the cumulants of the FCS as the limits [71]
S(ν) = lim
K→∞SK(ν) (3.7)
of the series
SK(ν) = XK m=1
s(ν)m Cm. (3.8)
For the Rényi entropies of integer order, the coecientss(ν)m are independent of the cut-o and read [140]
s(ν)m =
( (−1)m/2(2π)m2ζ[−m,(1+ν)/2]
(ν−1)νmm! m even,
0 m odd , (3.9)
whereζ(s, a) = P∞
n=0(n+a)−s is the generalized zeta function.
For non-interacting electrons, the FCS generally takes the form of a generalized binomial distribution [40, 41]. However, in some situations one may assume that the charge uctuations are essentially Gaussian, so that only the rst and second cumulants are non-zero. The entanglement entropy then becomes [69]
S ' π2
3 C2, (3.10)
having used the limiting value
K→∞lim a2(K) = π2
3 (3.11)
for the prefactor. Similarly, the Rényi entropies read
S(ν) 's(ν)2 C2 (3.12)
for nearly Gaussian uctuations.