Quantum Photovoltaic Effect in Double Quantum Dots
Abstract
We analyze the photovoltaic current through a double quantum dot system coupled to a high-quality driven microwave resonator. The conversion of photons in the resonator to electronic excitations produces a current flow even at zero bias across the leads of the double quantum dot system. We demonstrate that due to the quantum nature of the electromagnetic field in the resonator, the photovoltaic current exhibits a double peak dependence on the frequency of an external microwave source. The distance between the peaks is determined by the strength of interaction between photons in the resonator and electrons in the double quantum dot. The double peak structure disappears as strengths of relaxation processes increases, recovering a simple classical condition for maximal current when the microwave frequency is equal to the resonator frequency.
pacs:
73.23.-b, 73.63.Kv, 42.50.Hz, 73.50.PzI Introduction
The interaction of electrons in conductors with electromagnetic fields has long been considered within a classical picture of electromagnetic (EM) radiation. A widely–known example is the photon assisted tunneling (PAT) in double quantum dot (DQD) systems,van der Wiel et al. (2002) when the EM field brings an electron trapped at the ground state to an excited state and facilitates electron transfer. This classical description of the EM field breaks in high-quality microwave resonators based on superconducting transmission line geometry.Wallraff et al. (2004) Interaction of such EM fields with electronic devices require a quantum treatment known as the circuit quantum electrodynamics (cQED).Blais et al. (2004); You and Nori (2011)
Recently, several experimental groups studied systems consisting of a superconducting high quality resonator and a DQDFrey et al. (2012); Delbecq et al. (2011); Toida et al. (2012); Chorley et al. (2012); Petersson et al. (2012) or a voltage biased Cooper pair box.Astafiev et al. (2007) The coupling strength between a resonator photon mode and electron states in a DQD is characterized by the vacuum Rabi frequency with reported values in the range of Hz. These systems call for re-examination of the PAT by taking into account a quantum description of the EM field in terms of photon excitations. One may expect at least two important distinctions from the classical treatment: (1) the Lamb shift that renormalizes quantum states of electrons and photons; (2) spontaneous photon emission that breaks symmetry between absorption and emission processes and is important in systems with either a finite voltage bias between the leadsChildress et al. (2004); Sánchez et al. (2007); Jin et al. (2011) or an inhomogeneous temperature distribution.Vavilov and Stone (2006)
In this paper we study the photovoltaic current through a DQD coupled to a high-quality microwave resonator at zero bias across the DQD. The resonator is driven by external microwave source that populates a photon mode of the resonator, see Fig. 1. The photons excite electrons in the DQD and produce electric current even at zero bias, similar to the classical PAT case.van der Wiel et al. (2002); Stoof and Nazarov (1996); Pedersen and Büttiker (1998) We show that due to the coupling of electrons and photons, the current as a function of the source frequency has a multiple peak structure with splitting between the peaks determined by the coupling strength and reflects the Lamb shift of electronic energy states. We also demonstrate that the interaction-induced splitting is sensitive to the energy and phase relaxation rates in the DQD.
We note that the photovoltaic effect discussed here is a common phenomenon when the current in an electronic circuit is generated by out-of-equilibrium EM environment. Examples of this phenomenon include the current response of a DQD in the vicinity of the biased quantum point contactOuyang et al. (2010) or another circuit element out of equilibriumSilva and Kravtsov (2007) with the electronic system. However, because out-of-equilibrium photons of the environment have a broad spectrum, the generated current does not exhibit a resonant dependence on parameters of the system that we observe in a system of a single mode high quality resonator and a DQD.
Ii Model
We consider a DQD system with each dot connected to its individual electron reservoir at zero temperature and at zero bias between the reservoirs, see Fig. 1(b,c). The gate voltages of the DQD are adjusted near a triple point of its stability diagram.van der Wiel et al. (2002) To be specific, we choose a triple degeneracy point between , and electron states in the DQD and denote these states as , and , respectively. We model the system by the Hamiltonian , where describes states with an extra electron in the left or right dot, or :
(1) |
with being the electrostatic energy difference between the two states, and being the tunnel matrix element of an electron between the dots. The Pauli matrices are defined in the subspace of states and as and . A resonator driven by an external microwave source is described by the Hamiltonian
(2) |
with denoting the annihilation (creation) operators for microwave photons in the resonator, being the amplitude of the external drive of the resonator and () being frequency of the resonator (source). The interaction between the microwave field and the DQD system is represented byChildress et al. (2004)
(3) |
This interaction describes the shift of energy difference between states and due to the electric potential of the plunger gates defined by the microwave photon field. We assume that the photon field is distributed between the left and right plunger gates, see Fig. 1(b) and does not influence the source and drain voltage to avoid the rectification effects.Brouwer (2001); DiCarlo et al. (2003); Vavilov et al. (2005)
Further calculations are more convenient in the basis of the ground, , and excited, states of the Hamiltonian, Eq.(1):
(4) |
Here characterizes the hybridization of the or states. The energy splitting between the eigenstates can be tuned independently by varying and via dc gate voltages. We further eliminate the time-dependence in Hamiltonian Eq.(2) by applying unitary operator and utilize the rotating frame approximation to obtainChildress et al. (2004); Jin et al. (2011)
(5) | ||||
where characterizes the actual strength of the coupling between the microwave field and DQD states responsible for photon absorption or emission, the Pauli matrices , and are defined in terms of eigenstates of the electron Hamiltonian, Eq. (1).
We analyze the behavior of the system with Hamiltonian Eq. (5) in the presence of relaxation in electron and photon degrees of freedom by employing the Born-Markov master equation for the full density matrix
(6) |
The first term on the r.h.s. of Eq.(6) describes the unitary evolution of the system and the second term accounts for the dissipative processes in the resonator and DQD systemsOuyang et al. (2010)
(7) |
where is the Lindblad superoperator. The relaxation of the photon field in the resonator with rate is represented by and the electron relaxation from the excited state to the ground state with rate is represented by . The last two Lindblad superoperators account for the loading of the ground state and unloading of the excited state of the double quantum dot via electron tunneling in terms of operators and , respectively. The tunneling rates , , and are written in terms of tunneling rates in the basis of and states.
Note that in Eq.(6), the dynamics of state only appears via the tunneling terms involving and . These terms can be categorized by whether the empty state is loaded from the left or right lead with coefficients depending on projection of the eigenstates onto the left/right states, as shown in Fig. 1. In this picture,Ouyang et al. (2010) the photovoltaic current is given by
(8) |
in terms of the reduced density matrix , where we traced out photon degrees of freedom of the resonator. We also analyze the number of photons in the resonator,
(9) |
where we trace out both photon and electron degrees of freedom.
Iii Results
The average number of photons in the resonator, , and the dc component of photocurrent can be found using the steady state solution of the master equation, (6), with . We numerically find the full density matrix for a double quantum dot and photon field of the resonator in the Fock’s space using Quantum Optics ToolboxTan (1999) and QuTiPJohansson et al. (2012), both of which provide consistent results. The steady state solution for the density matrix defines the average number of photons , Eq. (9), and the photocurrent, Eq.(8).
Our choice of parameters is motivated by Ref. [Toida et al., 2012]. We choose the relaxation rate , the resonator relaxation rate , tunneling amplitude between the individual dots , the tunneling rate from a dot to a lead , and the resonator frequency GHz. We note that to keep the coupling constant finite, we have to take , since , Eq.(5), vanishes for . Below we fix MHz.
First, we investigate dependence of the photocurrent on the separation between energy levels in the double quantum dot, controlled by the electrostatic energy difference . We take frequency of microwave source to be equal to the resonator frequency, , and fix the drive amplitude . Dependence of the average number of photons in the resonator and the photocurrent on energy is presented in Fig. 2 for three values of the dephasing rate MHz. As the energy difference between the excited and ground states of the quantum dot goes through the resonance , we observe a significant suppression of the photon number in the resonator, see the top panel and the inset in Fig. 2. This is expected behavior because the DQD system enhances photon absorption in the resonator at . Absorbed photons cause transitions between the ground and excited electronic states. These electrons tunnel to the leads and generate electric current though the DQD. This current is shown in the lower panel of Fig. 2 and is peaked at or GHz, indicated by dashed vertical lines.
One feature in Fig. 2 is that the photon number is also reduced at zero bias , when the photovoltaic current vanishes. This suppression is a result of strong enhancement of the coupling constant at , resulting in stronger dissipation in the resonator and increase of off-resonant absorption rate. At the same time the photovoltaic current vanishes at due to cancellation between the two terms in Eq.(8).
The curves for the photon number and the current do not significantly change after the dephasing rate is introduced in addition to the energy relaxation rate . Dephasing smears the resonant condition for the photon absorption by the DQD and has two effects: (1) the number of photons increases a little near the resonance , see the inset in Fig. 2; (2) the resonant absorption of photons by electrons is suppressed resulting in reduction of the photocurrent. We note that in the case presented in Fig. 2 the first effect is stronger than the second effect and dephasing increases the magnitude of photocurrent for the case of fixed .
Next, we consider the case when the frequency of the microwave source, , is varied while the energy splitting of the DQD and the resonator frequency are fixed. The microwave radiation is mostly reflected when its frequency does not match the difference between energies of the resonator and DQD system defined by the Jaynes-Cummings spectrum:
(10) |
where is the detuning between the DQD and the resonator. We demonstrate that for DQD with weak energy and phase relaxations, this resonant admittance of the microwave source to the resonator results in the peak structure of the average photon number and the photocurrent.
In Fig. 3, we plot the average number of photons in the resonator and the photocurrent as a function of the drive frequency for and for the choice of other system parameters identical to those for curves in Fig. 2. At vanishing dephasing rate, , we observe a double peak feature in both photon number and photocurrent curves, see Fig. 3. These peaks at are defined by the level spacing of the Jaynes-Cummings Hamiltonian and are shown by vertical dashed lines in Fig. 3. The two peaks merge at as the dephasing rate increases and destroys quantum entanglement between photons and DQD states.
At finite detuning between the resonator and the DQD, , the eigenstates of the system become dominantly photon states or electron states of the DQD. As a result, the microwave source increases the number of photon excitations in the resonator when the microwave frequency is in resonance with the transition between the photon–like states, . But the source has a weak effect at the resonance with the electron-like states at frequency . We present the corresponding dependence of the photon number and the photocurrent in Fig. 4 for , and other parameters identical to those for in Figs. 2 and 3. We indeed observe one large peak in the photon number near the resonant condition for the dominantly photon state with energy while the photon number does not show significant enhancement near the second resonance, corresponding to the transition to the dominantly electronic state with energy . The photocurrent still exhibits double peak feature, but the peak corresponding to the photon resonance is higher, when the microwave drive produces a higher photon population.
We now consider a more idealistic regime of significantly reduced tunneling and relaxation rates , the drive amplitude MHz and . In this case additional resonances develop, see Fig. 5. These resonances correspond to excitations of several photons in the cavity by the microwave source. When the frequency of the source satisfies , the DQD-resonator system experiences transitions from the ground state to the energy state , cf. Ref. Blatt et al., 1995. These multiphoton transitions result in peaks of the average photon number and the magnitude of the photocurrent. Curves in Fig. 5 have three pairs of peaks at frequencies marked by vertical dashed lines for . We notice that for the average photon number is nearly the same, see the top panel in Fig. 5(a), while the photon distribution function is different, Fig. 5(b): at a non-zero develops for a probability that the resonator contains two photons. This difference in indicates that the microwave drive line does not match the resonator to produce a two photon occupation of the resonator at , but it matches the resonator to populate the state with the energy , which then decays to the lower energy states with .
Next, we investigate dependence of the photon number in the resonator and the magnitude of the photovoltaic current for different amplitudes of the drive. The above discussion was mostly focused on a resonator containing less than one photon. As the drive increases, the double peak feature evolves to a single peak at the drive frequency equal to the frequency of the resonator, . We interpret this cross-over as a signature of changed hierarchy of the terms in the system Hamiltonian. At weak drive, we have a JC Hamiltonian with its peculiar energy levels, Eq.(10), and the drive can be viewed as a weak probe testing the spectral structure of the coupled resonator and DQD system. Once the drive reaches the strength of the coupling, , a proper way to treat the system is to start with the Floquet–type statesPedersen and Büttiker (1998); Marthaler and Dykman (2006); Dykman (2012) of the driven resonator and then to take into account the interaction of these states with the DQD system as a perturbation. In this picture, the photon resonance happens at . The coupling is responsible for the formation of the broader “wings” in curves for the average photon number and the photocurrent. These wings are more pronounced in the photovoltaic current, which is entirely due to the coupling between resonator and DQD. This broad structure of the generated current as a function of the source frequency is preserved even at stronger drive. Thus, the shape of the photovoltaic curve might provide an experimental approach to quantify the strength of the JC coupling constant.
Iv Discussion and Conclusion
We analyzed the photovoltaic current through a DQD system at zero voltage bias between the leads. The double quantum dot interacts through its dipole moment to a quantized electromagnetic field of a high quality microwave resonator. The interaction is described by the Jaynes–Cummings Hamiltonian of a quantized electromagnetic field and a two level quantum system, represented by ground and excited electronic states of the double quantum dot. When a weak microwave radiation is applied to the resonator, the source acts as a spectral probe that causes excitation of the system when the energy difference between its eigenstates is equal to the photon energy of the source. If this resonance condition is satisfied, the microwave source populates the photon mode of the resonator and generates a direct current though the double dot system even at zero bias.
We demonstrated that at finite, but still low energy and phase relaxation rates of the DQD, both the average number of photons in the resonator and the photocurrent through the DQD have a double-peak structure as functions of the frequency of the microwave source. This double peak structure reflects an avoided crossing of the energy states of the DQD and the resonator photons due to the interaction between the two subsystems and is reminiscent of the Lamb shift by a single electromagnetic mode. We also found that in the limit if extremely weak relaxation rates of the DQD, multiphoton resonances develop when the energy difference between the states of the coupled system is a multiple of .
As energy and phase relaxation rates of the DQD increase, the peaks in the photon number and the photocurrent broaden and eventually merge in a single resonance peak at the frequency of the resonator. In this limit, the resonator photon mode and the DQD are no longer described as an entangled quantum system and the resonant condition for the interaction of the microwave source with the system corresponds to equal frequencies of the source and the resonator mode, .
At stronger microwave drive, frequency dependence of the average photon number in the resonator evolve from the Jaynes-Cummings double peaks at to a single peak at the resonator frequency . The single peak at is a result of multi-photon transitions at strong drive by the microwave source that all merge together due to finite width of multi-photon resonances. Similar evolution to a single peak occurs for the photocurrent response, although the photocurrent curve has a broader width as a function of the source frequency , this width corresponds to the strength of the coupling between the photon mode of the resonator and the DQD and may be used to characterize the strength of this coupling in experiments.
Acknowledgements.
We thank R. McDermott and J. Petta for fruitful discussions. The work was supported by NSF Grant No. DMR-1105178 and by the Donors of the American Chemical Society Petroleum Research Fund.References
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