Real-time dynamics in strongly correlated quantum-dot systems

Yong-Xi Cheng(程永喜), Zhen-Hua Li(李振华), Jian-Hua Wei(魏建华), and Hong-Gang Luo(罗洪刚)

1Department of Science,Taiyuan Institute of Technology,Taiyuan 030008,China

2Lanzhou Center for Theoretical Physics,Key Laboratory of Theoretical Physics of Gansu Province,and Key Laboratory of Quantum Theory and Applications of the Ministry of Education,Lanzhou University,Lanzhou 730000,China

3Beijing Computational Science Research Center,Beijing 100193,China 4Department of Physics,Renmin University of China,Beijing 100872,China

Keywords: quantum dots,mesoscopic transport,decoherence

1.Introduction

Quantum dots are small regions defined in a semiconductor material with a size of order 100 nm.[1]The wide range of novel physical phenomena of quantum dots lead to a very active and fruitful research topic, such as artificial atoms,coherent time-dependent effect of single-molecule magnets,[2,3]Kondo effect and non-Fermi-liquid behavior,[4]and bulk Kondo insulators.[5]Strongly correlated quantumdot systems are of great interests because of their fundamental physics as well as potential applications.[6]And strongly correlated electron materials exhibit fascinating collective behavior which has long challenged our understanding.[7]The notable phenomena of the strongly correlated quantum-dot systems are helpful,such as quantum criticality in heavy fermion systems,[8]Mott metal-insulator transition in transition metal oxides,[9]and high-temperature superconductivity in copper oxides.[10]While the prominent one is the dynamical properties of the strongly correlated quantum-dot systems, both of excited states near the Fermi energy and at highly excited energies.[7]

The practical importance of real-time dynamics in quantum-dot systems for quantum computing was emphasized by Elzermanet al.,[11]the temporal response to gate-voltage pulses for a single-shot is used to detect the spin configuration of a quantum dot in an external magnetic field.[11,12]Quantum dynamics is discussed in terms of quantum information theory, which indeed facilitates discussions between physicists, chemists, mathematicians, and quantum engineers.[13]The real-time dynamical property in quantum-dot systems is primely important for understanding the quantum transport through the nanodevices and successfully be used to track individual glycine receptors in the neuronal membrane of living cells in biology.[14]

Recent theoretical and experimental efforts aimed at observing and modeling nonequilibrium dynamical physics.The Keldysh technique[15]is succeeding in nonequilibrium dynamics.This approach to calculate the current bowing into and out of the interacting region treats the contacts as the systems separately in equilibrium in the distant past, possibly with different chemical potentials.[15-17]Jeremy Tayloret al.provided the first principle density functional theory for modeling quantum-dot dynamical properties.[18]Ping Zhang studied the dynamics of double quantum dots with the help of the Floquet formalism.The dynamical localization can be built up near the multi-photon resonances due to the structure exchange of the Floquet states at the avoided crossing.[19]However, the above mentioned methods are not capable enough to treat the strongly correlated effect accurately.The time-dependent density-matrix renormalizationgroup method (TD-DMRG) works well for one-dimensional quantum systems in finite size and short time, but is unsuitable for tackling long time scales own to the accumulated error proportional to the time elapsed.[20-23]Wilson’s numerical renormalization-group method is a prominent numerical tool for describing the equilibrium Kondo regime.[24,25]Furthermore,Anders and Schiller developed a time-dependent numerical renormalization-group approach(TD-NRG)to investigate the nonequilibrium dynamics of quantum-dot systems.[12]But there is also a deviation in occupancy from the new equilibrium state at the large time due to the difference between the time-evolved density matrix in a finite-size system and the equilibrium density matrix.[12]Overall,when considering both the electron-electron interaction of quantum-dot and the finite bandwidth of leads for practical cases, the accurate description of the long-time dynamical behaviors of quantum-dot systems is still an open question.

In this work, we study the real-time dynamics of quantum-dot systems by means of the hierarchical equations of motion approach(HEOM).[26-30]We consider respectively single quantum-dot system and serial coupling double quantum-dot system.The transient behavior of real-time dynamics and time-dependent occupancy response of a sudden change of gate voltage are presented.These results are closely relevant for experiments involving potentially important technological applications of quantum dots and quantum wires.[20]

The paper is organized as follows.In Section 2 we briefly review the HEOM approach, give the common formalism for real-time dynamics in quantum-dot systems, and present the relevant results for electron occupancy.Then we exhibit the time-dependent occupancy for the resonant-level model driven by a gate voltage,the results are consistent with by the exact analytical solution (EAS) of Keldysh formalism in Ref.[12].In Section 3 with taking into account the strongly correlated electron-electron interaction(U),firstly, we investigate the time-dependent occupancy in single quantum-dot systems with different factors, then the time-dependent occupancy in serial coupling double quantum-dot system.Distinct time evolution phenomena which can be realized experimentally are presented.In Section 4 we will give a summary of our work.

2.General formalism of HEOM in quantum-dot system

The hierarchical equations of motion approach (HEOM)is potentially useful for addressing quantum-dot systems especially for the interacting strong correlation systems.The outstanding characterizing both equilibrium and nonequilibrium properties achieved in our previous work are referred to Refs.[26,31-33].The HEOM approach has been employed to study dynamical properties, for instance, the dynamical Coulomb blockade and dynamical Kondo memory phenomena in quantum dots.[29,30]It is essential to adopt an appropriate truncated level to close the coupled equations.The numerical results are considered to be quantitatively accurate with increasing the truncated level and converge.In Ref.[31], it has been demonstrated that the HEOM approach achieves the same level of accuracy as the latest NRG method for the prediction of various dynamical properties at equilibrium and nonequilibrium.[34]Here,we solve the real-time dynamics problem and focus on the nonequilibrium dynamics of quantum-dot systems based on the HEOM.The local quantum dot constitutes the open system of primary interest, and the surrounding reservoirs of itinerant electrons are treated as environment.The total Hamiltonian for the quantum-dot systems

Now, to test our method, we consider a resonant-level model(RLM),describing the hybridization of a localized level with a band of spinless conduction electrons calculated in Ref.[12] by TD-NRG approach and EAS in the wide band limit.The total Hamiltonian of the system

We calculate the occupancynµ(t) =〈 ˆd†ˆd〉(t) of the RLM.It can be solved exactly in closed analytical form using the Keldysh formalism in the wide band limit.[12]

Figure 1 depicts the calculated time-dependent occupancynµ(t)at different temperatures by the HEOM approach in response to a sudden change in the energy of the level fromE0µ=0 toE1µ=-∆, and the converged tier level (L=4) is adopted.To compare the results of the EAS and TD-NRG approach described in Fig.2(b)in Ref.[12],we adopt the same parameters.Furthermore,we calculate the time-dependent occupancynµ(t) at seven different temperaturesT/∆=0.015,T/∆= 0.03,T/∆= 0.1,T/∆= 0.5,T/∆= 1,T/∆= 2,andT/∆=5.In comparison with the results in Ref.[12],we find that at short-time scale,all three methods(HEOM,EAS,and TD-NRG)give the perfect solution of the time-dependent occupancy.In particular,nµ(t →0+) coincides with the initial equilibrium state occupancy of RLM.The value of occupancy isnµ(0)=0.5 for all the temperatures at the time oft=0.While,at the large-time scale,there is a deviation from the new equilibrium occupancy between the EAS and the TDNRG approach.Anderset al.presented an explanation for the occupancy deviation.The long-time deviations innµ(t)using TD-NRG approach stem from a difference between the time-evolved density matrix in the finite logarithmic discrete state and the equilibrium density matrix ˆρwhen approachedΛ →1+.[12]

Fig.1.Time-dependent occupancy nµ(t) of the RLM versus time t calculated by the HEOM approach at different temperatures T/∆=0.015,0.03,0.1,0.5,1,2,5 for the log10 abscissas.The inset is the timedependent occupancy nµ(t) for the linear abscissas.The parameters adopted are E0µ =0,E1µ =-∆,and W =20∆.

Not only the short-time dynamics but also the long-time behavior is well described by our HEOM approach.The occupancynµ(t) of the new equilibrium state at the temperatureT/∆= 0.1,T/∆= 0.5,T/∆= 1, andT/∆= 5 are 0.75, 0.71, 0.65, and 0.55, those are the same values calculated by EAS.But the TD-NRG approach cannot get the same results.The occupancy deviation is developed at large-time scale.Those are greatly reduced by averaging over the differentz’s in TD-NRG approach.[12]Obviously,part of our calculations also cover the same parameters of Fig.2(b)in Ref.[12].The HEOM accurately reproduces the exact results by EAS at all the time scales at different temperatures.Our result exhibits the same dynamical behavior in short-time scale with EAS, and avoids the deviation in long-time scale appearing in TD-NRG.For more comparisons between those methods,please see our previous work in Ref.[35].Moreover,HEOM is an accurate and universal approach and is capable of addressing a variety of equilibrium and nonequilibrium, static and dynamical properties of strongly correlated quantum-dot systems.The method is essentially nonperturbative.In practical implements,an appropriate truncated level is taken to meet the limited computation resource.However, the high precision can be achieved by the convergence with increasing the truncation level.[35-37]

3.Occupancy of strongly correlated quantumdot systems

In this section we present two applications of the formalism developed above.Firstly, we investigate the timedependent occupancy with several factors in the strongly correlated single quantum-dot system by using HEOM approach,such as temperature and bandwidth of the leads.As the electron-electron interaction (U) is taken into account in the Hamiltonian of the device, the physical properties of realtime dynamics approach to the actual case.It is helpful to understand the Kondo effect, Mott metal-insulator, hightemperature superconductivity, and so on.To better understand and explore the real-time dynamical properties of the many-body systems,we also focus on the serial coupling double quantum-dot system case.The physical properties of dynamics under different temperatures and tunneling couplings between the two dots are presented.

3.1.Single quantum-dot system

We first consider the single quantum-dot systems modeled by the Hamiltonian

Here, ˆd†σand ˆdσdenote the creation and annihilation operators for spin-up and spin-down electrons on the dot,Uis the electron-electron interaction, and ˆnσ=↑,↓= ˆd†σˆdσis the number operator.To investigate the relaxation in the spin,we consider the following stepwise change in the energy of the levelEµ(t)=θ(-t)E0µ+θ(t)E1µ.We begin with a degenerate spin stateE0µ=∆/2.At the timet=0, the energy of the level is shifted toE1µ=-U/2-∆/2.The resulting time evolution of the quantum-dot occupancy isnµ(t)=〈ˆn↑+ˆn↓〉(t).[12]

Figure 2 depicts the occupancynµ(t) of the single quantum-dot system for the different values of electronelectron interactionU.With the electron-electron interactionUincreases, the quantum-dot system changes into a strongly correlated regime.The initial occupancy of the system at the timet=0 decreases with the electron-electron interactionU.The time scales for the relaxation of system are clearly visible.Thenµ(t) equilibrates on a time scaletch∝1/∆, and the Rabi-type oscillations are developed for|E1µ|>∆.The Rabi-type oscillations are enhanced by the electron-electron interactionU,and the time when the occupancy oscillates becomes earlier with the system correlating strongly.Fort ≫tch,the quantum-dot system will get a new steady state,and the occupancy must saturate at its new equilibrium valuenµ(t)=1(Fig.2(a)).[12]To emphasize the real-time dynamical behavior of the quantum-dot system, we slice thenµ(t) curve forU=18∆in Fig.2(b).We find that the occupancynµ(t) of the quantum-dot system firstly shows a distinct oscillating behavior and finally reaches a new equilibrium steady state value scaling with the time.

Fig.2.Time-dependent occupancy of the single quantum-dot system versus time t at different values of U/∆= 2, 4, 6, 8, 10, 12, 18.The parameters adopted are T =0.015∆, W =20∆, E0µ =∆/2, and E1µ =-U/2-∆/2.

To compare with the RLM, we calculate the timedependent occupancynµ(t)of the single quantum-dot system for different temperatures.Figure 3 plots the dynamical behavior of the occupancy under the same electron-electron interactionU=18∆and bandwidthW=20∆.Unlike the RLM,the occupancy of the strongly correlated single quantum-dot system oscillates at the low temperature(see the lines ofT=0.015∆,T=0.03∆,T=0.1∆,T=0.5∆in Fig.3).Because the strongly correlated system changes into the Kondo regime at low temperatures, the Kondo memory effects are expected to be more prominent.So the occupancynµ(t) at the initial equilibrium state shows a nonmonotonic transition behavior.The occupancynµ(t)firstly decreases and then increases with increasing temperature.At high temperaturesT= 1∆andT=2∆,the oscillation vanishes and the occupancy increases linearly with the timetbefore the system gets the new equilibrium state.In RLM,the oscillation behavior is absent and the occupancy increases linearly with the timetfor all the temperatures.The values of the occupancy in new equilibrium state decrease monotonously with the temperature increasing(Fig.1).While in the strongly correlated single quantum-dot system, the values of the occupancy in new equilibrium state saturate almost atnµ=1.

Fig.3.Time-dependent occupancy of the single quantum-dot system versus time t at different values of T/∆=0.015, 0.03, 0.1, 0.5, 1,2.The parameters adopted are W =20∆,E0µ =∆/2,E1µ =-U/2-∆/2,and U =18∆.

The real-time dynamical properties of the quantum-dot system are presented by EAS approach,only applicable in the case ofU=0, with the calculation of the wideband limit approximation.Here we can study the dynamical properties of the quantum-dot system in a finite bandwidth.In Fig.4, we show the evolution of the time-dependent occupancynµ(t)-tcurves at several bandwidthsWin Kondo regime.We find that the narrow bandwidth enhances the Rabi-type oscillation of the occupancy.At large bandwidth, the occupancynµ(t) oscillates several times and gets a new equilibrium state quickly after the time scales for the relaxationtch∝1/∆.At the narrower bandwidth,the occupancynµ(t)displays very sharp oscillations and owns a larger oscillation frequency.This behavior mainly results from the bandwidth enhancement of the capacitive contributions from the accumulation and depletion of electrons layering on either side of the quantum-dot system leads.[35]The narrow bandwidth does not have sufficient time to follow the stepwise change in the energy of the level.The temporal coherence of electrons tunneling through the quantum-dot system leads to the oscillations more obviously with decreasing the bandwidth.Moreover, the initial equilibrium state occupancy of the single quantum-dot system also increases with the bandwidth.Those outstanding behaviors of real-time dynamics in narrow bandwidth cannot be acquired by TD-NRG with wideband limit approximation.

Summarizing Figs.2-4, one can conclude that the Rabitype oscillation of the occupancy is strongly dependent on the electron-electron interactionU,temperatureT,and bandwidthW.The Rabi-type oscillation can be enhanced by strong electron-electron interactionU, low temperatureT, and narrow bandwidthW.

Fig.4.Time-dependent occupancy of the single quantum-dot system versus time t at different bandwidths W/∆=20, 12, 4, 2.The parameters adopted are T =0.015∆,E0µ =∆/2,E1µ =-U/2-∆/2,and U =18∆.

3.2.Double quantum-dot system

Double quantum-dot systems can be considered to some extent as artificial molecules,it is useful for the study of many novel phenomena involving the strong Coulomb interaction,antiferromagnetic spin coupling, and time-dependent coherence for realizing solid state quantum bits.[19,38-40]The coherent dynamics of a single charge qubit in a double quantumdot system is discussed with full one-qubit manipulation.[13]Several experiments on photon-assisted tunneling through the double quantum-dot system have been carried out.[19]Here,we focus on the real-time dynamical properties through the serial coupling double quantum-dot system.The Hamiltonian for the system

Here,we firstly focus on the different temperaturesTfor non-interaction and then finite interactions between the two dots, respectively.Figure 5 depicts the time-dependent occupancy of the serial coupling double quantum-dot system versus timetat different values ofT/∆=0.015,0.1,0.5,1 for the electron-electron interaction between the two dotsU12=0(a) andU12=4∆(b).We find that the Rabi-type oscillation of the occupancy is distinct forU12=0 at low temperatures.This oscillating behavior is suppressed by increasing temperatures.This Rabi-type oscillation vanishes and the occupancy grows monotonically and linearly with time at high temperatures.The above transition behaviors are equal to the single quantum-dot system case.Moreover,the occupancy of the serial coupling double quantum-dot system in new equilibrium state reachesnµ=2 as the double value of the single quantumdot system(Fig.5(a)).However,the oscillating behavior of the occupancy vanishes atU12=4∆.The occupancy increases monotonically with time for all the temperatures (Fig.5(b)).Here, the coherence bonding state forming between the two quantum dots dominates the dynamical transport behavior of the serial coupling double quantum-dot system.It causes the oscillation of the occupancy disappearing atU12=4∆.

Fig.5.Time-dependent occupancy of the serial coupling double quantum-dot system versus time t at different values of T/∆=0.015,0.1,0.5,1 for the electron-electron interaction between the two dots U12 = 0 (a) and U12 = 4∆(b).The parameters adopted are W =18∆,E0µ =∆/2,E1µ =-U/2-∆/2,and U1=U2=-2Eµ.

To further understand how the electron-electron interaction between the two dotsU12affects the real-time dynamical properties of the serial coupling double quantum-dot system, we then calculate the time-dependent occupancynµ(t)of the serial coupling double quantum-dot system with differentU12.Figure 6 presents the numerical result of occupancynµ(t) for the serial coupling double quantum-dot system.We find that the Rabi-type oscillations of the occupancynµ(t)are strongly dependent on the electron-electron interaction between the two dotsU12.The oscillating behavior of the serial coupling double quantum-dot system is suppressed by the electron-electron interaction between the two dotsU12.AtU12=0, the occupancy owns a distinct oscillating behavior associated with a larger amplitude.This behavior becomes faintish with increasing the electron-electron interaction between the two dotsU12.TheU12-dependent oscillating behavior of the serial coupling double quantum-dot system can be understood as follows.When a stepwise changes the energy level, it accompanies the change of the electron numbers.However, with increasing the electron-electron interactionU12, the variation of the electron number at either of the dots is suppressed.So the Rabi-type oscillation of the serial coupling double quantum-dot system becomes attenuated with increasingU12.Moreover, the frequency of Rabi-type oscillations decreases sharply fromU12=0 toU12=2∆.AtU12=2∆,the occupancynµ(t)only owns a wriggle behavior as timetincreases.

Fig.6.Time-dependent occupancy nµ(t)-t curves of the serial coupling double quantum-dot system with different electron-electron interactions between the two dots U12.The parameters adopted are T =0.015∆, E0µ =∆/2, E1µ =-U/2-∆/2, U1 =U2 =-2Eµ, and W =2∆.

Fig.7.Time-dependent occupancy nµ(t)-t curves of the serial coupling double quantum-dot system with different bandwidths W for the electron-electron interaction between the two dots U12 = 0 (a) and U12 =4∆(b).The parameters adopted are T =0.015∆, E0µ =∆/2,E1µ =-U/2-∆/2,and U1=U2=-2Eµ.

Finally,we explore the oscillating behavior of occupancy of the serial coupling double quantum-dot system with different bandwidths of the leadsWat two values of the electronelectron interaction between the two dotsU12.Figure 7 shows the time-dependent occupancynµ(t) of the serial coupling double quantum-dot system with different bandwidthsWfor the electron-electron interaction between the two dotsU12=0(a) andU12= 4∆(b), respectively.We find that the distinct oscillating behavior of occupancy at narrower bandwidth becomes attenuated with increasing the bandwidth for bothU12=0 andU12=4∆.For the electron-electron interaction between the two dotsU12=0,the frequency of Rabi-type oscillations decreases regularly.However,the oscillation shows an irregular transition behavior forU12=4∆.Here, both the bandwidthWand the electron-electron interaction between the two dotsU12manipulate the oscillating behavior of the serial coupling double quantum-dot system.It leads that the frequency of Rabi-type oscillations decreases sharply with increasing the bandwidth.

4.Summary

In summary,we calculated accurately the time-dependent occupancy of the strongly correlated single quantum-dot system and serial coupling double quantum-dot system subject to a sudden change of gate voltage.The Rabi-type oscillation of the strongly correlated single quantum-dot system can be enhanced by strong electron-electron interaction, low temperature, and narrow bandwidth.Moreover, the timedependent occupancy of the serial coupling double quantumdot system shows an irregular transition behavior, when both the bandwidth and the electron-electron interaction between the two dots manipulate the oscillating behavior.The oscillating behavior of the serial coupling double quantum-dot system is suppressed by the electron-electron interaction between the two dots.Those characteristics of strongly correlated quantum-dot systems may be observed in experiments.And we believe our results are helpful for investigating the real-time dynamical properties of driven quantum many-body systems and will lead to significant new insights in the future.

Acknowledgements

Project supported by the National Natural Science Foundation of China(Grant Nos.11804245,11747098,11774418,12247101,and 12047501),the Scientific and Technologial Innovation Programs of Higher Education Institutions of Shanxi Province, China (Grant No.2021L534), and the Fund from the Ministry of Science and Technology of China (Grant No.2022YFA1402704).