Unification of electromagnetic noise and Luttinger liquid via a quantum dot

Abstract

We investigate the effect of dissipation on a small quantum dot (resonant level) tunnel-coupled to a chiral Luttinger liquid (LL) with the LL parameter K. The dissipation stems from the coupling of the dot to an electric environment, being characterized by the resistance R, via Coulomb interactions. We show that this problem can be mapped onto a Caldeira-Leggett model where the (ohmic) bath of harmonic oscillators is governed by the effective dissipation strength α = (2K) −1 withK −1 = K −1 + 2R/RK and RK = h/e 2 the quantum of resistance. Experimental consequences are discussed and the limit K = 1/2 + is thoroughly studied at small R/RK through the spin-boson-fermion model. PACS numbers: 73.23.Hk, 71.10.Pm, 72.70.+m A quantum dot can be viewed as a simple artificial atom exhibiting charge quantization [1], and its charge can now be measured with a very high accuracy with the aid of an electrometer based on a single-electron transistor [2]. The coupling of the quantum dot to a macroscopic reservoir of electrons inevitably produces quantum charge fluctuations on the dot. This generic phenomenon has been vividly investigated theoretically both in the case of a large metallic box with a very dense spectrum [3] and in the opposite limit of a two-level system [4]. The reservoir of electrons may be a two-dimensional (2D) Fermi-liquid lead [3,5] or a 1D structure [6,7] [e.g., a fractional quantum Hall edge state (FQHES)or a quantum wire] where interacting electrons form TomonagaLuttinger liquid (LL). Some recent endeavors have been accomplished by Cedraschi et al. [4] and by one of us [8] to understand the role of dissipation-coming from the capacitive coupling of the dot to an electric environment with an ohmic resistance-on the charge quantization of a quantum dot, but with the limitation of free electrons in the reservoir lead. In this Letter, we explore dissipation effects on the small quantum dot (resonant level) coupled to a chiral LL (CLL) which has been previously introduced by Furusaki and Matveev [6]. We seek to provide a unified picture of the role of interactions in the onechannel conductor and of the zero-point fluctuations of the electric environment generalizing the case of a single junction [9]. Of interest to us is to understand the nature of the quantum phases emerging through the dissipative mesoscopic structure shown in Fig. 1 and hence to discuss physical implications for the occupation probability on the quantum dot. We highlight that the physics explored here is sufficiently appealing to experimentalists to carry out activities similar to those already existing on superconducting qubits capacitively coupled to lossy transmission lines in GaAs/AlGaAs heterostructures [10].The quantum dot of interest in Fig. 1 is small enough such that we can only restrict ourselves to the highestoccupied level. This leads to a two-level system [4]. The gate voltage V g is fixed such that the two states in which the level is occupied (|1 ) or not (|0 ) are almost degener- ate. We can thus resort to an orbital s...

Proposed Method for Tunneling Spectroscopy with Ohmic Dissipation Using Resistive Electrodes: A Possible Majorana Filter

“…Since the correlation function of the phase shows the similar power law decay to the chiral bosonic field: he ÀiðtÞ e ið0Þ i $ t À2r and he ÀiÈ ðx¼0;tÞ e iÈ ðx¼0;0Þ i $ t À1 , we can combine the two bosonic fields and introduce a new field [32,40,41]: È ðxÞ ¼ ffiffiffi g p ½È ðxÞ þ ðxÞ with g ¼ 1=ð1 þ 2rÞ, which satisfies he ÀiÈ ðx¼0;tÞ e iÈ ðx¼0;0Þ i $ t À1 . Note that only ðx ¼ 0Þ ¼ has the physical meaning (i.e., phase fluctuation), and ðx Þ 0Þ are auxiliary fields.…”

Nonequilibrium occupation number and charge susceptibility of a resonance level close to a dissipative quantum phase transition

“…First, through similar bosonization and refermionization procedures as in equilibrium, [3][4][5][6] we map our model to an equivalent anisotropic Kondo model in an effective magnetic field h with the effective left L and right R Fermi-liquid leads. 12 The effective Kondo model takes the form Note that ␥ → Ϯ V / 2 near the transition ͑␣ ‫ء‬ → 1 / 2 or ␣ → 1͒ where the above mapping is exact.…”

“…In recent years, there has been a growing interest in QPTs in nanosystems. [3][4][5][6][7][8][9][10][11] Very recently, QPTs have been extended to nonequilibrium correlated nanosystems [12][13][14] where little is known regarding nonequilibrium transport near the transitions. A generic example 12 is the transport through a dissipative resonance level ͑spinless quantum dot͒ at a finite bias voltage where dissipative bosonic bath ͑noise͒ coming from the environment in the leads gives rise to QPT in transport between a conducting ͑delocalized͒ phase where resonant tunneling dominates and an insulating ͑localized͒ phase where the dissipation prevails.…”

“…A generic example 12 is the transport through a dissipative resonance level ͑spinless quantum dot͒ at a finite bias voltage where dissipative bosonic bath ͑noise͒ coming from the environment in the leads gives rise to QPT in transport between a conducting ͑delocalized͒ phase where resonant tunneling dominates and an insulating ͑localized͒ phase where the dissipation prevails. Though dissipative QPTs have been investigated intensively in various systems, 3,4,7,8,12,[15][16][17][18] much of the attention has been focused on equilibrium properties while very little is known on the nonequilibrium properties. The bias voltage V plays a very different role as the temperature T in equilibrium systems as the voltage-induced decoherence behaves very differently from the decoherence at finite temperature, [12][13][14]19,20 leading to exotic transport properties near the QPT compared to that in equilibrium at finite temperatures.…”

Driven quantum dot coupled to a fractional quantum Hall edge

“…In this paper we discuss a system motivated by recent experiments and related theoretical work which study singleparticle electron emitters [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] . The model is comprised of a single chiral FQHE edge state coupled to a quantum dot via a quantum point contact (QPC), and we explore the dynamics of this model in a non-equilibrium setup.…”