Correlation between lasing and transport properties in a quantum dot-resonator system

Abstract

We study a double quantum dot system coherently coupled to an electromagnetic
resonator. By suitably biasing the system, a population inversion can be
created between the dot levels. The resulting lasing state exists within a
narrow resonance window, where the transport current correlates with the lasing
state. It allows probing the lasing state via a current measurement. Moreover,
the resulting narrow current peak opens perspective for applications of the
setup for high resolution measurements.

Full text

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arXiv:1109.1166v1 [cond-mat.mes-hall] 6 Sep 2011
Correlation between lasing and transport properties
in a quantum dot-resonator system
Pei-Qing Jin1, Michael Marthaler1, Jared H. Cole1,2, Maximilian
K¨ opke3, J¨ urgen Weis3, Alexander Shnirman4,5, and Gerd Sch¨ on1,5
1Institut f¨ ur Theoretische Festk¨ orperphysik, Karlsruhe Institute of Technology, 76128
Karlsruhe, Germany
2Applied Physics, School of Applied Sciences, RMIT University, Melbourne 3001, Australia
3Max-Planck-Institut f¨ ur Festk¨ orperforschung, D-70569 Stuttgart, Germany
4Institut f¨ ur Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, 76128
Karlsruhe, Germany
5DFG Center for Functional Nanostructures (CFN), Karlsruhe Institute of Technology, 76128
Karlsruhe, Germany
E-mail: pei-quing.jin@kit.edu
Abstract.
resonator.
the dot levels. The resulting lasing state exists within a narrow resonance window, where the
transport current correlates with the lasing state. It allows probing the lasing state via a current
measurement. Moreover, the resulting narrow current peak opens perspective for applications
of the setup for high resolution measurements.
We study a double quantum dot system coherently coupled to an electromagnetic
By suitably biasing the system, a population inversion can be created between
Circuit quantum electrodynamics (CQED) setups with superconducting qubits coupled to
a superconducting resonator provide possibilities to explore quantum optics effects in new
parameter regimes [1]. Recently lasing and cooling effects were demonstrated in such systems
[2, 3, 4]. The strong coupling regime achieved in these setups revealed qualitatively novel
phenomena. An example is the non-monotonous behavior of the linewidth of the emission
spectrum in a single qubit maser, which is influenced by quantum noise in a characteristic way
[5, 6].
Here we propose a different CQED setup where a double quantum dot is coupled to a
superconducting resonator in a geometry indicated in Fig. 1. The double quantum dot is biased
such that a single electron occupies the lowest empty orbital in the left or right dot. These two
relevant states are denoted as pure charge states |1,0? and |0,1? with an energy difference ǫ.
A coherent interdot tunneling with strength t is assumed to couple the two pure charge states.
The resulting eigenstates are mixtures of pure charge states with mixing angle θ = arctan(t/ǫ).
Transitions between these eigenstates are allowed due to an electrical dipole interaction with the
resonator. Under the rotating wave approximation, the Hamiltonian for the coupled system in
the eigenbasis of the two-level system is given by the standard Jaynes-Cumming Hamiltonian
Hsys=?ω0
2
σz+ ?ωra†a + ?g(a†σ−+ aσ+). (1)
Here ω0=√ǫ2+ t2/? and ωrdenote the frequencies of the two-level system and the resonator,

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Figure 1. Left panel: Illustration of a double quantum dot-resonator circuit. The dot is placed
at a maximum of the electric field of a coplanar waveguide (CPW). Right panel: Tunneling
sequence in the double dot system. The electro-chemical potentials µL(1,0) and µR(0,1) of the
two dots are assumed to be arranged as indicated.
respectively, where a detuning ∆ = ω0− ωris allowed, and a (a†) represents the annihilation
(creation) operator of photons in the resonator.
The dynamics of the coupled dot-resonator system is studied in the frame of a master equation
for the reduced density matrix ρ. Within the conventional Born-Markovian approximation, the
master equation is given by
˙ ρ=−i
?
?Hsys,ρ?+
?
i
Γi
2
?
2LiρL†
i− L†
iLiρ − ρL†
iLi
?
. (2)
The dissipative dynamics is described in Lindblad form with operators Li. For the decay of
resonator, we adopt the standard form Lr= a with rate Γr= κ. For relaxation and decoherence
of the two-level system, the corresponding Lindblad operators are given by L↓= σ−with rate Γ↓
and Lϕ= σzwith rate Γ∗
ϕ. Throughout the paper we consider low temperatures with vanishing
thermal photon number and excitation rates. The incoherent tunneling between electrodes and
quantum dots are also included (see below).
To achieve the population inversion for lasing we assume the system to be biased such that
only the electro-chemical potentials of the states |1,0? and |0,1? lie within the bias window. At
low temperatures compared to the charging energy, an electron can only tunnel into the dot
system from the left electrode to the left dot. It leads to a transition from the state |0,0? to
|1,0?. This process is described by a Lindblad operator LL= |1,0??0,0|. Similarly, an electron
can tunnel out into the right lead, creating a transition from |0,1? to |0,0?, which is described
by LR= |0,0??0,1|. For simplicity, we assume these two processes having the same tunneling
rate Γ. A non-equilibrium state is achieved with enhanced population in the state |1,0? with
higher energy as compared to the state |0,1?. This effect persists in the eigenbasis. The resulting
population inversion in the absence of relaxation between dot levels, becomes
τ0=
4cosθ
3 + cos(2θ). (3)
As the system approaches the degeneracy point with θ = π/2, the pure charge states are coupled
strongly by the interdot tunneling, and the population inversion decreases.
A current flows through the double dot when the tunneling cycle |0,0? → |1,0? → |0,1? →
|0,0? is completed. We evaluate the current using I = e?
refers to the eigenstates of the two-level system as well as the state |0,0?, and Γi→j denotes
i,jΓi→j?i|ρst|i?, where the index i

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?1.0
?0.50.0 0.5 1.01.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Ε ?units of ? Ωr?
I ?units of e??
?a?
?0.005
0
0.005
0
1
2
3
4
5
6
7
? ?units of Ωr?
?b?
F
?n??5
I??0.1 e??
Figure 2. (a) Current as a function of the energy difference ǫ. (b) Average photon number
?n?, Fano factor F and transport current I as functions of the detuning. In both panels, we
choose the interdot tunneling strength t = 0.3 ?ωr, incoherent tunneling rate Γ = 10−3ωr, decay
rate κ = 10−5ωr, relaxation and decoherence rates Γ↓= Γ∗
g = 3 × 10−4ωr.
ϕ= 10−4ωr, and coupling strength
the transition rate from state |i? to |j?. The transport current at low temperature (vanishing
thermal photons) is plotted in Fig. 2 as a function of the energy difference ǫ. At the degeneracy
point ǫ = 0, a broad peak shows up due to the coherent interdot tunneling, which width is given
by the tunneling rate t (here t ≫ ?Γ) [7, 8]. With respect to the degeneracy point, the current
exhibits asymmetric behavior. On the absorption side (ǫ < 0), an electron that tunnels into the
left dot is blocked in the state |1,0? since no photon is present to provide the energy for tunneling
to the right dot. Therefore on the absorption side, only the residue from the broad current peak
remains. On the emission side (ǫ > 0), a second current peak arises when the two-level system
becomes resonant with the oscillator. For realistic values of the parameters, this current peak
is much sharper. It correlates with a lasing state within a narrow resonance window with width
W ≈ 2g?Γ/κ − 1 for small θ [9], since a photon is generated in the resonator when an electron
tunnels between the two dots. Besides, the relaxation of the two-level system, which opens up
an incoherent channel, increases the overall current on the emission side.
The lasing state is characterized by the average photon number ?n? as well as the Fano
factor F ≡ (?n2? − ?n?2)/?n?. As indicated in Fig. 2, at low temperature, the photon number
vanishes for large detuning, since the quantum dot system does not interact effectively with the
resonator. The system is in the non-lasing regime and the Fano factor can be approximated by
F ≃ ?n? + 1. When moving towards the resonance, the system undergoes a lasing transition,
accompanied by a sharp increase in the photon number, and an enhanced Fano factor since the
amplitude fluctuations increase. At resonance, the photon number reaches a maximum, and the
Fano factor is near 1, indicating the system is in a coherent state.
To study experimentally the predicted lasing effect, it is most suitable to process a double-
quantum-dot system with in-situ tunable parameters which allow controlling the electron
numbers and energy levels within the quantum dots but also the tunnel couplings towards
the leads and/or between the quantum dots. A double quantum dot system based on a 2D
electron system (2DES) which is embedded in a GaAs/(AlGa)As heterostructure close to the
heterostructure surface offers such properties.
negative voltages to structured metal electrodes on top of the heterostructure surface, or by
etching grooves into the heterostructure to the depth of the 2DES and thereby dividing the
2DES into regions acting as leads, gates, and quantum dots. The second approach is more
suitable in our case as the CPW resonator can be directly deposited on the heterostructure
without being affected by normal conducting metal electrodes which might cause energy loss for
the resonator. By removing the 2DES in most areas of the resonators’s mode volume – except
The 2DES is structured by either applying

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for the region of the quantum dot system in the center of maximum field of the resonator – the
interaction of the electrically dissipative 2DES with the resonator field is diminished.
In such an approach, the typical geometrical diameter of a quantum dot is 0.5 to 0.7 µm,
leading – due to electrostatic depletion from the etched grooves – to an effective quantum dot
diameter of about 0.2 to 0.4 µm. The respective single-electron charging energies (e2/2C) are
then in the range of 0.5 to 1 meV. Excitations in such quantum dots can be characterized by
electrical transport spectroscopy and are found at energies which are typically 1/10 to 1/3 of
the single-electron charging energy. A disadvantage of such structures might be that the tunable
tunnel barriers are energetically shallow and spatially broad.
The proposal here relies on a double-quantum dot where a single electron jumps from one dot
to the other. The respective energy difference ǫ for the electron sitting on either quantum dot
is tunable by applying gate voltages. The respective electrical dipole change d which couples to
the resonator field is about the distance between both quantum dot centers times the elementray
charge, i.e. here about d = 0.6µm×e. For an electric field strength of E = 0.2V·m−1- typical
of such a CPW resonator, we find a maximum coupling factor of g = d · E/? ≈ 20MHz.
Achieving the lasing action puts constraints on parameters, namely, a relatively strong
coupling compared to dissipations [6],
g >
?κΓϕ
2τ0
, (4)
where Γϕdenotes the total rate accounting for all sources of dissipation of the two-level system.
For small θ it reduces to Γϕ≃ Γ↓/2 + Γ∗
substrates with κ = 10−4ωr have been reported [10], two orders of magnitude larger than
achieved on other substrates. This value of κ would be sufficient to obtain lasing, if Γϕcan be
kept in the range of 10 MHz.
On the other hand, the lifetime of an electron inside the double-quantum dot system has to
be long enough to allow for stimulated photon emission while making the transition from one dot
to the other. The respective tunnel rates can be tuned to low value. However, we want to detect
currents, which under optimum conditions requires current above 1 pA, i.e. the tunneling rates
have a lower limit given by 1/τ < I/e ≈ 0.6MHz. This would suffice to detect the lasing peak
of magnitude about 2eΓ/3 [9]. Given sufficiently high coupling it should also be possible [11] to
observe the change in photon number characteristic of lasing.
ϕ+ Γ/4. To present date, CPW resonators on GaAs
Acknowledgments
We acknowledge helpful discussion with S. Andr´ e and A. Romito, as well as the financial support
from the Baden-W¨ urttemberg Stiftung via the Kompetenznetz Funktionelle Nanostrukturen.
References
[1] Schoelkopf R and Girvin S 2008 Nature 451 664
[2] Astafiev O, Inomata K, Niskanen A O, Yamamoto T, Pashkin Yu A, Nakamura Y and Tsai J S 2007 Nature
449, 588
[3] Grajcar M, van der Ploeg S H W, Izmalkov A, Il’ichev E, Meyer A G, Fedorov A, Shnirman A and Sch¨ on G
2008 Nature Physics 4, 612
[4] Marthaler M, Sch¨ on G and Shnirman A 2008 Phys. Rev. Lett. 101, 147001
[5] Andr´ e S, Brosco V, Shnirman A and Sch¨ on G 2009 Phys. Rev. A 79, 053848
[6] Andr´ e S, Jin P Q, Brosco V, Cole J H, Romito A, Shnirman A and Sch¨ on G 2010 Phys. Rev. A 82, 053802
[7] van der Vaart N C, Godijn S F, Nazarov Yu V, Harmans C J P M, Mooij J E, Molenkamp L W and Foxon
C T 1995 Phys. Rev. Lett. 74, 4702
[8] Stoof T H and Nazarov Yu V 1996 Phys. Rev. B 53, 1050
[9] Jin P Q, Marthaler M, Cole J H, Shnirman A and Sch¨ on G arXiv:1103.5051 [cond-mat.mes-hall]
[10] Frey T, Leek P J, Beck M, Ensslin K, Wallraff A and Ihn T arXiv:1104.3535 [cond-mat.mes-hall]
[11] Schuster D I, Houck A A, Schreier J A, Wallraff A and Gambetta J M 2007 Nature 445, 515