Electron transport of a driven three-level system in an asymmetric double quantum dot
irradiated by an external field
Y. Y. Liao and D. S. Chuu*
Department of Electrophysics, National Chiao-Tung University, Hsinchu 300, Taiwan Y. N. Chen
Department of Physics and National Center for Theoretical Sciences, National Cheng-Kung University, Tainan 701, Taiwan
共Received 19 April 2006; revised manuscript received 30 December 2006; published 27 March 2007兲
Electron tunneling through a three-level system in an asymmetric double quantum dot irradiated by an external field is investigated. For a resonant external field, two symmetric peaks occur in the current spectrum. If the field frequency is detuned, unequal contributions from two channels lead to two asymmetric peaks with population inversion, which can be observed with increasing Rabi frequency. On the other hand, as the ground states in two dots are equal, a suppression of current occurs around the resonant frequency. In contrast, an enhanced behavior is found for the case of unequal ground states.
DOI:10.1103/PhysRevB.75.125325 PACS number共s兲: 73.21.La, 73.63.⫺b, 73.23.Hk
I. INTRODUCTION
Due to the zero dimensionality, quantized energy levels, and stable states, transport properties of the electrons in quantum dots have been studied extensively.1,2With the
ad-vance of nanotechnologies, quantum dots can be laterally fabricated from a two-dimensional electron gas in a hetero-structure. Together with the gate technology, the leads and single共multi-兲 quantum dot are able to form a quantum de-vice. Since the features of the dots are controllable, the study of external influences on quantum dots becomes an impor-tant topic.3,4
Recently, quantum dot systems in the presence of time-varying external fields manifest some interesting effects ranging from photon-assisted tunneling5 to electron
pumping.6In a recent experiment, the transport spectroscopy
has been measured in coupled double quantum dots under microwave fields.5The photon-assisted resonances are found
due to a modulated gate voltage. The phenomenon involves the emission or absorption of a microwave photon. In addi-tion to electron tunneling, other systems based on time-dependent influences also give rise to some promising phys-ics and applications.7–11
On the theoretical side, many transport studies under driv-ing fields were focused on two-level systems in sdriv-ingle or double quantum dot.12–17The transport property is basically
related to the energy difference between two levels. Although many investigations are focused on the artificial two-level systems, to the best of our knowledge, however, electron tunneling through a three-level system still receives little attention.18In the present work, we thus propose to study the
electron tunneling through a three-level system in a double quantum dot. By applying an external field on the device, two peaks are found in the current spectrum. Besides, the positions of two peaks are varied with different strengths of the external field. A crossover from the three-level to the two-level system is further pointed out, depending on the frequency of the external field. By analyzing the contribu-tions from the two channels in the right dot, a population inversion can be observed in the system. Furthermore, we
also study the effect of field frequency on the transport in the case of various energy differences between the ground states in two dots. It is found that the current shows a suppressed or an enhanced behavior.
II. MODEL
With the advances of nanotechnologies, the double-dot 共charge qubit兲 system in the Coulomb blockade regime can now be fabricated and measured as demonstrated in Refs.19
and20. The charging energies of the two quantum dots, left and right, are⬃4 and ⬃1 meV, respectively, and the average spacing between single-particle states is ⬃0.5 and ⬃0.25 meV.19According to the experimental parameters, we
thus consider a three-level system defined in a double quan-tum dot system, as shown in Fig.1. The large charging en-ergies enable one to reasonably assume that there is maxi-mally one extra electron in this double-dot system.19–22
Furthermore, since the single-particle spacing for the left and right dots is different共⬃0.25 meV兲, it is plausible to neglect the excited state in the left dot; i.e., when a continuous field 共with energy close to 0.25 meV兲 irradiates on the system, it only gives rise to the contribution of the first excited state of
FIG. 1. Schematic view of a three-level system which consists of the ground state in the left dot and the ground state and first excited state in the right dot in a double quantum dot device. An external field irradiates on the device and leads to the transition between two states in the right dot.
the right dot. We further estimate the transition between the ground state of the left dot and the excited state of the right dot. It is found that this contribution is much smaller than the contribution between the ground and the excited states of the right dot. In order to simplify the system, we reasonably concentrate on the transition between two states in the right dot. For the tunneling between two dots, the electron is only allowed to tunnel between two ground states. In this case, the effective Hilbert space of the electronic system can be de-fined by four states: empty共no electron exists in both dots兲, left共one electron in the ground state of left dot兲, right 共one electron in the ground state of right dot兲, and excited states 共one electron in the excited state of right dot兲, corresponding to 兩0典=兩NL, NR, NE典, 兩L典=兩NL+ 1 , NR, NE典, 兩R典=兩NL, NR + 1 , NE典, and 兩E典=兩NL, NR, NE+ 1典, respectively. The total Hamiltonian of the system is
H = Hres+ Hdot+ HV+ HT+ Hep. 共1兲 The first term describes the electron reservoir contribu-tions, Hres=
兺
k苸L k L ck†ck+兺
k苸R k R dk†dk, 共2兲where ck共ck†兲 is the annihilation 共creation兲 operator in the left lead 共L兲 with wave vector k and dk共dk
†兲 is the annihilation 共creation兲 operator for the right lead 共R兲. The term Hdot de-scribes the contributions of three states in the double-dot system,
Hdot=LnˆL+RnˆR+EnˆE, 共3兲 where the energy levelsL,R, and E represent the ground state in the left dot, the ground state, and the first excited state in the right dot, respectively. The operators of the three states are given by nˆL=兩L典具L兩, nˆR=兩R典具R兩, and nˆE=兩E典具E兩. The dot-lead coupling can be written as
HV=
兺
k Vk L ck†sˆL+兺
k Vk R dk†sˆR+兺
k Vk E dk†sˆE+ H.c., 共4兲 with the operators sˆL=兩0典具L兩, sˆR=兩0典具R兩, and sˆE=兩0典具E兩, and the tunneling matrix elements Vk␣for ␣共=L, R, and E兲. The term HTdescribes the tunneling between the ground states in two dots,HT= Tc共Pˆ + Pˆ†兲, 共5兲 where the operator Pˆ 共Pˆ†兲 is defined by 兩L典具R兩 共兩R典具L兩兲 and the tunnel matrix element Tc determines the strength of the tunneling process. In the dipole and rotating-wave approxi-mations, the last term Hep which describes the interaction between electron and external field in the right dot can be expressed as
Hep= −
␥
2共Qˆe
−it+ Qˆ†eit兲, 共6兲 where␥ is the Rabi frequency,is the field frequency, and the operator Qˆ 共Qˆ†兲 denotes 兩E典具R兩 共兩R典具E兩兲. The Rabi fre-quency relates to the field strength and the electric dipole moment for the transition兩R典↔兩E典.23,24
An analytical expression for the stationary current can be solved from the master equation.17One can obtain an
equa-tion of moequa-tion for the time-dependent expectaequa-tion values of the operators nˆL, nˆR, nˆE, Pˆ , and Qˆ. After the Laplace trans-formation 共e.g., nL共z兲=兰0⬁dt e−zt具nˆL典t兲, the corresponding equations can be written as
nL共z兲 = − i Tc z +⌫L 关P共z兲 − P†共z兲兴 + ⌫L z +⌫L 关1/z − nR共z兲 − nE共z兲兴, nR共z兲 = i Tc z +⌫R 关P共z兲 − P†共z兲兴 + i ␥ 2共z + ⌫R兲 ⫻关Q共z + i兲 − Q†共z − i兲兴, nE共z兲 = − i ␥ 2共z + ⌫R兲 关Q共z + i兲 − Q†共z − i兲兴, P共z兲 = − i2Tc关2共z − i+ i⌬L兲 + ⌫R兴 A 关nL共z兲 − nR共z兲兴 +2␥Tc A Q †共z − i兲, Q共z兲 = − i␥关2共z − i⌬L兲 + ⌫R兴 2B 关nE共z − i兲 − nR共z − i兲兴 +␥Tc B P †共z − i兲, 共7兲 where A =关2共z − i+ i⌬L兲 + ⌫R兴关2共z − i⌬兲 + ⌫R兴 +␥2, B =关2共z − i⌬L兲 + ⌫R兴关z − i⌬R + ⌫R兴 + 2Tc2,
with the parameters ⌬=L−R, ⌬L=E−L, and ⌬R=E −R, respectively. The tunneling rates between the reservoirs and dots are assumed to be energy independent: ⌫␣ = 2兺k兩Vk␣兩2␦共␣−k
L/R兲 with ␣ 共=L, R, and E兲. Note that, according to the calculations, some particular relations be-tween the states are attributed to the influence of the external field. For example, it is found that a collective effect on the right dot can be generated in the presence of the driving field. This gives rise to a contribution of the transition be-tween the ground state of the left dot and the excited state of the right dot. These contributions are further incorporated into Eq. 共7兲. We can solve Equation 共7兲 algebraically and
subsequently obtain the stationary current共in units of e兲 from the tunneling between two dots,
I = iTc共P − P†兲t→⬁. 共8兲
To simplify the parameters of the system, the tunneling rates are assumed to be identical共⌫L=⌫R=⌫E兲. According to Refs.
19 and 20, the tunneling rate 共⌫兲, tunneling coupling 共Tc兲, and energy spacing共E−R兲 in this work are set equal to the values of 1eV, 1eV, and 0.25 meV, respectively.
III. RESULTS AND DISCUSSIONS
We first consider that the field frequency is in resonance 共⌬=−⌬R=0兲 and apply the external voltages to vary the energy difference between the ground states.19–22The current
can be written as
I = 4Tc
2⌫ 12Tc
2+ 4共⌬兲2+⌫2+ f共␥,⌬兲, 共9兲 where f共␥,⌬兲 is directly dependent on the Rabi frequency. For a two-level system共␥= 0兲, the current is
I = 4Tc
2⌫ 12Tc2+ 4共⌬兲2+⌫2
共10兲 and shows a maximum response at ⌬=0 共see Fig. 2兲.15
However, as a resonant field is applied to the device, the current shows an interesting behavior. With the increase of the field strengths, two symmetric peaks occur in the current spectrum. Compared to the case of␥= 0, the maximum cur-rent does not locate at the point共⌬=0兲. The positions of the peaks are mainly dependent on the related parameters such as the tunneling rates, tunnel coupling, and Rabi frequency. For the case of strong field, the Rabi frequency becomes dominant. Accordingly, it is found that the interval between the two peaks is roughly equal to the value of the Rabi frequency.
We also analyze the components of the current in this device. For the right dot, the ground and first excited states can contribute to the transport, as shown in Fig. 1. In the stationary case, Eq.共8兲 is equivalent to the contributions of
two states in the right dot. The current can be rewritten as
I = IR+ IE, 共11兲
IR=⌫nR, 共12兲
IE=⌫nE, 共13兲
where nR and nE are the populations in the right dot. From the inset in Fig. 2, we find that electron tunneling through two channels behaves similarly and contributes equally to the current.
In order to study the influence of the external field on the transport, in Fig. 3 we illustrate the curve of the Rabi-frequency-dependent current. For simplicity, the conditions are chosen to be⌬=0 and ⌬= 0. The current can be writ-ten as I = 4Tc 2⌫ 12Tc2+⌫2+ f共␥兲 , 共14兲 f共␥兲 =␥2␥ 2− 6T c 2+⌫2 ␥2+ 2T c 2+⌫2. 共15兲 If the Rabi frequency ␥ is zero, the current is 4Tc2⌫/共12Tc2 +⌫2兲. As the frequency ␥ increases, a crossover from en-hanced behavior to suppressed behavior in the transport spectrum is found due to the competition among the Rabi frequency ␥, tunneling coupling Tc, and tunneling rate ⌫ 关see Eq. 共15兲兴. In particular, it is found that a population inversion is observed in the inset, approximately
correspond-ing to the tendency of the current. To analyze the properties of the device in detail, we simplify the parameters by letting
Tc=⌫ and the populations of two states in the right dot can be subsequently taken as nR= 12⌫4 ␥4+ 8␥2⌫2+ 39⌫4, 共16兲 nE= 4␥2⌫2 ␥4+ 8␥2⌫2+ 39⌫4. 共17兲 In the limit of ␥→0, the population in the ground state mainly contributes to the current. However, as one increases
␥, there exists a crossing point when population inversion occurs, as can be seen from the inset of Fig.3. This is
be-FIG. 2.共Color online兲 Current as a function of energy difference ⌬ between two ground states for different Rabi frequencies. The inset shows the currents IR共dashed curve兲 and IE共dotted curve兲 for
␥=5⌫.
FIG. 3. Dependence of the current on Rabi frequency, corre-sponding to the populations nRand nE共inset兲. The conditions are
cause the population of the excited states first increases and then decreases with a rate of␥−2 关Eq. 共17兲兴, which is slower than␥−4 for the population in ground state关Eq. 共16兲兴.
Instead of studying the strength of the field, let us now turn our attention to the detuned field frequency. In Fig. 4, the current shows two asymmetric peaks. Though this may be similar to the result of Ref.25, our system with the res-ervoirs shows some different consequences. If the detuning is further increased, the main 共larger兲 peak is close to the value ⌬=0, while another one is far away from the main peak and deeply suppressed. This reflects a crossover from a three-level to a two-level system. In particular, as can be seen in the inset, the current 共solid curve兲 consists of two components IR 共dashed curve兲 and IE 共dotted curve兲, and both components contribute different degrees to the peaks of current. The main共secondary兲 peak mainly results from the contribution of current IR 共IE兲, respectively, i.e., the popula-tions behave inversely. The important feature is a large asymmetry in the magnitudes of the populations nR and nE contributed to the current peaks, e.g., 4:1 for the main one. To describe the appearance, one can expect that the external field establishes a particular relationship among the states. The distributions of populations are sensitive to the related parameters in the double-dot system. Under the condition ⌬⫽0, compared to the symmetric current 共Fig.2兲, the
elec-tron transferred among these states shows an unbalanced be-havior such that two channels in the right dot unequally con-tribute to the current peaks.
Figure5shows the dependence of the current on the de-tuning for various energy differences between two ground states. At⌬=0, a symmetric and antiresonant behavior ap-pears in the transport spectrum共red dashed curve兲. The cur-rent is greatly suppressed and the maximum response is lo-cated at resonant frequency 共⌬= 0兲. By altering the field frequencies, the ratio of the maximum to minimum current is about 8:1. This indicates that the operation of the external field can drastically influence the transport. When the energy
difference is further increased, it is found that an asymmetric and enhanced current occurs. In addition, the profile is squeezed and the location of maximum current is no longer fixed. This can be understood from the splitting and shifting of energy states modified by strong driving field in the right dot. For example, when ⌬=5⌫ 共black solid curve兲, the maximum current is approximately located at the resonant frequency 共⌬= 0兲, and the energy shift from the external field 共Rabi splitting兲 is roughly equal to the value of 5⌫. Therefore, one can utilize a suitable field to effectively con-trol the transport, such as a drastic drop 共enhancement兲 in current for the case of⌬=0 共5⌫兲. The highly sensitive re-sponse in the current spectrum might be very useful for the purpose of being a switch.
A few remarks about the comparison of our model with the related studies should be addressed here. For population inversion, the degree of the inversion in our work is manipu-lated by the strength of external field, while in Ref. 26 the inversion is achieved for the case of strong interdot Coulomb repulsion. As for the switching phenomenon, Chan et al.27
investigated a double-dot working as a switch through strong capacitive coupling provided by a floating interdot capacitor. Furthermore, Ono et al.28 showed that the Pauli spin
block-ade can be used to block current in a double dot. Different from the switching systems in Refs.27 and 28, our results suggest that the three-level system driven by an external field can advantageously play the role of switching in electron transport with a flexible characteristic via the control of the external field.
In order to realize our model, we suggest the experimental system studied by Fujisawa et al.19 An important
require-ment is the asymmetric double-dot structure. Although the work of Fujisawa et al. concentrated on two-level system, it is easily generalized to a three-level case with an adequate external field,29,30 i.e., an additional channel is opened to
contribute to the transport. Based on the successful studies and advanced nanofabrication technologies, we believe our model can be experimentally verified under proper arrangements.
FIG. 4.共Color online兲 Current as a function of energy difference ⌬ for different nonresonant fields 共⌬兲 and fixed Rabi frequency 共␥=5⌫兲. The inset shows that the total current I 共solid curve兲 is composed of two channels in the right dot: the electron tunneling out through the ground level IR 共dashed curve兲 and first excited level IE共dotted curve兲 for ⌬=4⌫.
FIG. 5.共Color online兲 Current as a function of frequency differ-ence⌬ 共=−⌬R兲 for different energy differences ⌬ between two ground states with fixed␥=10⌫.
IV. CONCLUSION
The transport of a three-level system under the influence of the external field is studied. When a resonant field irradi-ates on the double-dot device, the current shows two sym-metric peaks. For the case of strong field, the Rabi frequency dominates the interval between two peaks. In contrast, two asymmetric peaks display in a nonresonant field. When the detuning is increased, we find a crossover from the three-level to the two-three-level system. The population inversion can be observed by varying the frequency or strength of the ex-ternal field. Moreover, we also study the frequency-dependent current by modulating the energy difference
be-tween the ground states. It is clearly shown that a suppressed 共enhanced兲 behavior occurs due to the interplay among the states and external field.
ACKNOWLEDGMENTS
The authors gratefully acknowledge Rafael Sanchez and the anonymous referees for their useful comments and sug-gestions. This work is supported partially by the National Center for Theoretical Sciences and the National Science Council, Taiwan under Grant Nos. NSC 94-2112-M009-024 and NSC 95-2119-M-009-030.
*Electronic address: dschuu@mail.nctu.edu.tw
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