Project acronym

DYMESYS

Project title in Romanian

Dinamica in timp real in sisteme mesoscopice puternic corelate

Project title in English

Real-time dynamics in strongly correlated mesoscopic systems

Domains of expertise to which the project belongs

Physics

Keywords

Spintronics, transport in quantum dots / mesoscopic physics

Project duration

36 months  (January 2012 -  December 2014)

 

RESULTS OBTAINED SO FAR:

 

Preprints:

  

Finite-frequency-dependent noise of a quantum dot in a magnetic field

C. P. Moca, P. Simon, Chung-Hou Chung, G. Zarand

arxiv:1312.4648

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Published papers:
 

Emergent SU(4) Kondo physics in spin-charge-entangled double quantum dot

A. J. Keller, A. Amash, I. Weymann, C.P. Moca, I. G. Rau, J. A. Katine, H. Shtrikman, G. Zarand and G. Goldhaber-Gordon

Nature Physics 10, 145  – Published February 2014

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Finite-frequency thermoelectric response in strongly correlated quantum dots

Razvan Chirla and Cătălin Paşcu Moca

Phys. Rev. B 89, 045132 – Published 24 January 2014

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Fermi liquid theory of resonant spin pumping

C. P. Moca, A. Alex, A. Shnirman, and G. Zarand

Phys. Rev. B 88, 241404 – Published 10 December 2013

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Probing the Rashba effect via the induced magnetization around a Kondo impurity

R. Chirla, C. P. Moca, and I. Weymann

 Phys. Rev. B 87, 245133 – Published 27 June 2013

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SU(3) Anderson impurity model: A numerical renormalization group approach exploiting non-Abelian symmetries

Cătălin Paşcu Moca, Arne Alex, Jan von Delft, and Gergely Zaránd

Phys. Rev. B 86, 195128 – Published 19 November 2012

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Measurement of Quantum Noise in a Carbon Nanotube Quantum Dot in the Kondo Regime

J. Basset, A. Yu. Kasumov, C. P. Moca, G. Zaránd, P. Simon, H. Bouchiat, and R. Deblock

Phys. Rev. Lett. 108, 046802 – Published 24 January 2012

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 INVITED LECTURES, SEMINARS, TALKS, CONFERENCES

 

 

September 2013: Conference participation: "New trends in nanophysics and solar energy conversion", National Institute of Material Physics, Bucharest, Romania. (Presentations made by Catalin Pascu Moca and Razvan Chirla)

 

June 2013: Conference participation: "BIOSUN, Biomimetics sensing using nano-objects", National Institute of Material Physics, Bucharest, Romania. (Catalin Pascu Moca and Razvan Chirla)

 

May 2013: Conference participation: "Artifical atoms: from Quantum physics to applications", Budapest, Hungary (Catalin Pascu Moca)

 

March 2013: "Quantum noise in the Kondo regime", Seminar at the Department of Physics, Budapest University of Technology and Economics, Budapest, Hungary (Catalin Pascu Moca)

 

September 2012: “Transport in correlated quantum dots”, part of 3 lectures given at the Cargèse summer school on mesoscopic physics. (Pascal Simon)

 

May 2012: "From shot noise to quantum noise", invited talk, Ortvay Kollokvium, Budapest (Gergely Zarand)

 

May 2012: “Non-equilibrium frequency dependent noise in a quantum dot”, Talk at the International workshop “Statistical Physics and Low Dimensional Systems”, Nancy, France. (Pascal Simon)

 

February 2012: "Frequency dependent noise in a carbon nanotube quantum dot", Seminar at the Department of Theoretical Physics, Budapest University of Technology and Econonics, Budapest, Hungary. (Catalin Pascu Moca)

 

October 2011: “Non-equilibrium frequency-dependent noise through a quantum dot : A real time functional renormalization group approach”, Talk at the International workshop on``Charge and heat dynamics in nano-systems”, Orsay, France. (Pascal Simon)

 

October 2011: "Quantum noise of a Carbon nanotube ", invited talked, SFB 658 Symposium on Transport through molecules, Berlin (Gergely Zarand)

  

PEOPLE INVOLVED:

Romanian side:

 

Prof. Catalin Pascu Moca (Oradea, Romania)

Prof. Gergely Zarand (Budapest, Hungary)

CS II. Valeriu Moldoveanu (INFM, Bucuresti, Romania)

Dr.  Razvan Chirla (Oradea, Romania)

Adrian Roman (Oradea, Romania)

 

 

French side:

 

Prof. Pascal Simon (Orsay, Paris, France)

Dr. Richard Deblock (Orsay, Paris, France)

Dr. Dhananjay Dhokarh (Postdoc position from 01/04/2013 to 30/09/2013, 6 months)

 

 

 

DESCRIPTION OF THE PROJECT :

 

A. ABSTRACT :

The recent development in fabrication, and operating in a controlled manner of nanoelectronic devices with a few hundred angstroms scale or below, are likely to provide our future technology and serve as basic tools for storing information, quantum computation or spin manipulation. Understanding how these nanometric devices work represents a major challenge for today’s theoretical and experimental physics. To reach this goal we need first to figure out the fundamental issues that govern their behavior and therefore to provide a detailed theory of correlations and transport in atomic scale and mesoscopic structures. More specifically, in order to find efficient ways of manipulating and controlling the spin currents, we must understand the microscopic processes that lead to spin relaxation and dephasing, and eventually their interplay with interactions.

In the present project, our main purpose is to understand non-equilibrium transport through strongly correlated mesoscopic systems. To achieve this goal we shall exploit the expertise of two research partners. The first one, «Laboratoire de Physique des Solides» in Orsay, involves two complementary teams, a theoretical one (led by Pascal Simon) and an experimental one (led by Richard Deblock). The second partner, «University of Oradea» in Romania involves a theoretical team (led by Cătălin Pascu Moca). We plan to study non-equilibrium transport in mesoscopic systems subject to strong interactions, by combining the power of quantum field theoretical methods and numerical analysis with the experimental probation of the theoretical findings. Our research groups will develop new theoretical methods that will enable us to study and understand out of equilibrium, charge and spin transport through these nanometer scale devices. We plan to construct a real time renormalization group scheme, and then use it to investigate frequency dependent quantities that characterize transport under non-equilibrium conditions. Here we have in mind a thorough analysis of charge/spin ac-conductance and noise, scattering rates, relaxation effects and many other transport properties under non-equilibrium conditions. We shall also address the «quantum quench» problem in Kondo correlated systems, i.e. the implications on transport and the modifications in the response functions of a sudden change in the system Hamiltonian. To fulfill this goal, we first plan to carry an ambitious project by developing and extending the currently available numerical renormalization group code, the «Flexible-DMNRG», by incorporating time dependent processes.

On the experimental side we are planning to engineer high frequency quantum detectors for noise measurements in quantum dots in the Kondo regime made of carbon nanotubes. Quantum quench problem shall be investigated experimentally also, by using pulsed voltage probes. It will allow us to monitor different quantities, such as the conductance or noise subject to a quench in one of the system parameters.

We strongly believe that our approach is an optimal one for attacking real-time dynamics problems and that the interplay between the experiment and theory is the strength of the present proposal. More than that, our experimental findings will motivate and guide our theoretical developments along the way.

 

 

B. SCIENTIFIC AND TECHNICAL PROGRAMME, PROJECT ORGANISATION:

 

A) General out-of-equilibrium dynamics and transport through strongly correlated mesoscopic systems.

The main purpose of this topic is to build a real time functional renormalization group (FRG) scheme on the Keldysh contour and to study the finite-frequency transport properties and the current noise through Kondo systems (quantum dots) in the non-equilibrium regime. Probably the best known theory that goes beyond perturbation theory is the "poor man's scaling" approach of Anderson. The method was generalized to non-equilibrium problems for transport at finite voltages by A. Rosch et al. [Rosch2003]. In Ref. [Moca2010a], we propose a different approach, which is based on a real time functional renormalization group formalism. The renormalization group equations for the vertices are obtained by expanding the interaction action S and rescaling a cutoff parameter. Then an integro-differential equation is obtained for the vertices, which becomes relatively simple in Fourier space.

Our method reproduces the scaling equations of Rosch et al. [Rosch2003] for the vertex function. However, we have found that the current vertex also develops a non-trivial non-local structure in time, governed by a new set of RG equations. Such structure of the current vertex turns out to be unavoidable to guarantee current conservations and is necessary to calculate the finite frequency current noise in a controlled manner. Solving this set of RG equations, the complete frequency and temperature dependent noise spectrum through the dot can be analyzed. Our approach is valid at any frequency, voltage V, and temperature T provided that : max{eV, kBT } > kBTK.

We shall address the followings issues along this line:

A1) Non-equilibrium frequency dependent conductance, and the absorption /emission noise; Our primary purpose is to compute the current-current correlation functions. To do that, we first define the left and right current operators from the equation of motion in terms of the vertex matrices. It is useful that for any operator that we are interested in (the current operator in this particular case) to introduce a corresponding generating function from which the current-current correlation function is generated. Then the derivation of the correlator can be done in a systematic way, order by order in the perturbation theory on the Keldysh contour. The current operator is a non-local in time object, characterized by a kernel (current vertex), that satisfies a scaling equation similar to the one for the frequency dependent couplings. Next, the current-current correlator can be related to the lesser/greater noise which is also related to, the more accessible experimentally, absorption/emission noise. At the same time, the ac-conductance at a finite bias can be extracted from the noise expression through non-equilibrium fluctuation-dissipation theorem [Safi2009].

A2) Elastic/inelastic scattering rates in terms of the T-matrix; We shall focus on the scattering of conduction electrons off the Kondo impurities. A useful quantity in scattering problems is the so-called T-matrix and it is defined by the perturbative expansion of the conduction electron Green’s function. In terms of this quantity we can define scattering rates. Technically, this requires the calculation of the composite-Fermion correlator, which can be carried out through the introduction of a new non-local generating field, similar to the one for the current operator. The total scattering rate is related to the spectral representation of the T-matrix while the elastic scattering rate can be related to the full T-matrix. The difference between the two quantities gives the inelastic contribution. Besides providing information of the scattering rates, the T-matrix derived in this way can be related to the differential conductance for a finite voltage, and therefore, besides the noise itself, it gives a second route to evaluate the ac-conductance.

A3) Investigating the effects of external magnetic fields field on different dynamic correlations; In the presence of the magnetic field, the coupling vertex becomes strongly anisotropic, and the scaling equations suffer changes. It was shown in Ref [Rosch2003] that in the presence of external magnetic field the renormalized couplings show a singular behavior at energies ±|eV±B|. More than that, the mere presence of a magnetic field provides a separate source of dephasing. We shall analyze all these effects on the ac-conductance as well as on the noise and scattering rates.

A4) Transient response; We shall address the problem of transient regime, namely, the short-time dependent current response in one of the leads when a voltage is applied to other lead. This analysis can be formulated in terms of the frequency dependence response functions.

A5) Relaxation effects; So far we have considered that spin relaxation is due entirely to the interacting part of the Hamiltonian. On the other hand the spin dynamics is strongly affected by the environment. Since each nucleus carries spins, one of the most important sources of relaxation is the hyperfine interaction between the electron and the nuclear spin. This leads to a dephasing time of the order of 10ns or larger. Other mechanisms that may lead to dephasing, is the coupling to the phononic modes due to spin-orbit interaction or polaronic dephasing processes due to coherent acoustic phonon generation, in general, characterized by similar dephasing times. We shall estimate the change in noise spectrum due to the presence of a generic external dephasing mechanism characterized by a finite dephasing time. This question is rather crucial in order to efficiently compare theory with the experiments.

A6) Spin related emission/absorption noise and related issues; With the real-time FRG for the charge transport at hand, we can easily generalize it, and look for spin dependent quantities. Since the scaling equations for the vertex couplings do not change much, the corresponding correlations function can be straightforwardly derived. In this way we provide a route to analyze the full frequency dependence of the spin current correlation functions (spin noise) and the spin-resolved ac-conductance of quantum dots subject to a spin bias in the Kondo regime.

A7) Two level Anderson model with spin-orbit coupling; We would like to generalize the FRG scheme to more elaborate quantum impurity models. The double quantum dot with spin-orbit interaction (SOI) is probably one of the most interesting choices, because in the presence of SOI, spin is no longer a good quantum number, but a Kramers doublet remains as time reversal symmetry is conserved. Then, a Kondo effect in the presence of a finite gate voltage is possible provided that there is an odd number of electrons in the quantum dot. We shall therefore generalize the FRG scheme and address the effects of the spin-orbit interaction on the noise spectrum in a double quantum dot.

B) Experimental investigation of absorption/emission noise.

B1) Fabrication of quantum detectors. For experiments on Kondo physics, the characteristics of the high frequency noise detector depend on the system considered. For quantum dots made of carbon nanotubes the Kondo temperature are typically in the range of few kelvins. One may thus need to develop detectors working at higher frequency than the one made in aluminium (frequency range 5-100 GHz), by using material with a higher superconducting gap, like niobium.

grafic 1

B2) Measurements on Carbon nanotube Quantum dots in the Kondo regime; The detection of high frequency noise of a quantum dot in the Kondo regime coupled to a Josephson junction will rely on the measurement of the photo-assisted tunnelling (PAT) current of quasiparticles across the junction (Fig.1). This PAT current is related to the noise generated by the dot. By measuring the difference in the I(V) characteristics of the junction when the dot is biased and when it is not, we plan to measure the PAT current and extract the corresponding noise spectrum of the quantum dot by using a numerical deconvolution procedure. This procedure will be repeated for different bias voltages and gate voltages so as to explore the whole stability diagram of the dot. Depending on the bias condition of the detector, the measurement can be sensitive to the emission noise or the absorption noise.

B3) Optimizing the coupling between the carbon nanotube and the quantum detector; To optimize the coupling between the carbon nanotube quantum dot and facilitate the extraction of the current noise from the measurement of the PAT current of the detector, we plan to use a resonant circuit, as the one used in [Basset2010], to select the frequency coupled to the noise detector (Fig. 2). Consequently we will have measurement of the noise only at the resonance frequencies of the coupling circuit but for these frequencies the coupling should be bigger than the one obtained with on chip capacitances and resistances [Deblock2003, Billangeon2006, Billangeon2007].

The design of the coupling resonant circuit will have to be optimized to provide a good coupling, i.e. a high quality factor, at frequencies of interest for the physics of carbon nanotubes in the Kondo regime. We plan to test different design and materials (aluminium, niobium). There are some specific issues concerning the fabrication of the samples combining noise source and detector: since carbon nanotubes are obtained by CVD from magnetic catalyst deposited on the substrate, the Josephson junctions, which will be fabricated afterwards, need to be properly aligned so as not to come in contact with the catalysts.

B4) Interpretation of the data; The interpretation of the experimental data will certainly require and motivate some further theoretical developments. One of the expected difficulties may be to correctly take into account relaxation and decoherence effect induced by the bias voltage and the environment in general (see task A5). These effects are known to strongly reduce and weaken the Kondo resonance [Franceschi2002].

C) Quantum quenches in Kondo systems

We shall address the problem of real time dynamics in quantum impurity systems under an abrupt change in the system Hamiltonian, and study the implications on transport and the modifications in the dynamical correlation functions. Most of the theoretical work so far was focused on understanding the evolution of static transport quantities, such as the conductance, while the spectral properties of the system as well as the dynamical quantities were most of the time not investigated.

From a technical point of view a multitude of methods were used to address the quantum fluctuations problem under non-equilibrium conditions, such as the time dependent density matrix renormalization group (TD-DMRG) method [Schmitteckert2008, Schollwock2004], time-dependent numerical renormalization group (TD-NRG) [Anders2005], or the quantum Monte Carlo for non-equilibrium [Millis2009], to name a few.

We shall address the followings issues along this line:

C1) Construction of a TD-NRG scheme; Our primary purpose here is to construct a time-dependent numerical renormalization group scheme for the evaluation of both dynamical and static correlations under sudden changes of the system Hamiltonian, and to implement it into the Flexible-DMNRG package. This approach must be able to explicitly use the symmetries of the system.

C2) Spectral properties of a Kondo system under non-equilibrium conditions; Once the code is available, we shall investigate the non-equilibrium spectral properties and the time evolution of static quantities characterizing the Kondo systems. Here we have in mind a thorough investigation of both Kondo and Anderson models, when one of the system parameters, such as the exchange coupling J, or the external magnetic field B, in the case of the Kondo model, or the on-site energy E0, the Coulomb interaction U, or the broadening parameter Γ, in the case of Anderson model is suddenly changed. As we discussed in section A2, the T-matrix is an important quantity, since it provides information on experimentally measurable quantities such as the ac-conductance. Therefore, we shall address the effect of the non-equilibrium on this quantity as well. Here we shall investigate the large/small frequency limit of the T-matrix, and the implicit changes in the scattering rates and the dynamical conductance.

C3) Fermi liquid to Non-Fermi liquid crossover; In the case of a single-channel Kondo (1CK) problem, the system behaves always like a Fermi liquid (FL) at temperatures lower than the characteristic Kondo temperature TK. On the other hand, where there are two screening channels, as in the case of two-channel Kondo (2CK) problem, it is possible to have either a FL-like ground state, when one screening dominates over the other one, or a non-Fermi liquid (NFL) ground state, when the screenings are equal. We shall address the problem of a quantum quench in one of the channels and investigate the crossover from a FL to a NRL ground state in the 2CK model.

C4) Experimental investigation of quantum quenches in correlated quantum dots; The dynamics of the Kondo effect can be probed experimentally by making a sudden change in the gate voltage or the bias voltage. It can be done with the help of pulse voltage sources. The signature of the time dependant parameter can then be monitored on different quantities. The easiest measurement is to probe the DC conductance of the quantum dot. Noise measurement can also be done and benefit from the relatively high bandwidth of the noise detector and coupling circuit detailed in part B3.

 

 

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