The interface with spin defects in hexagonal boron nitride has recently become a promising platform and has shown great potential in a wide range of quantum technologies. Varieties of spin properties of $V_B^-$ defects in hexagonal boron nitride (hBN) have been researched widely and deeply, like their structure and coherent control. However, little is known about the influence of off-axis magnetic fields on the coherence properties of $V_B^-$ defects in hBN. Here, by using the optically detected magnetic resonance (ODMR) spectroscopy, we systematically investigated the variations in ODMR resonance frequencies under different transverse and longitudinal external magnetic field, respectively. In addition, we measured the ODMR spectra under off-axis magnetic fields of constant strength but various angles, and observed that the splitting of the resonance frequencies decreases as the angle increases, aligning with our theoretical calculation based on the Hamiltonian, from which we come up with a solution of detecting the off-axis magnetic field angle. Through Rabi oscillation measurements, we found that the off-axis magnetic field suppresses the spin coherence time. These results are crucial for optimizing $V_B^-$ defects in hBN, establishing their significance as robust quantum sensors for quantum information processing and magnetic sensing in varied environments.
This research employs the Kraus representation and Sz.-Nagy dilation theorem to model a three-level quantum heat on quantum circuits, investigating its dynamic evolution and thermodynamic performance. The feasibility of the dynamic model is validated by tracking the changes of population. On the basis of reinforcement learning algorithm, the optimal cycle of the quantum heat engine for maximal average power is proposed and verified by the thermodynamic model. The stability of quantum circuit simulations is scrutinized through a comparative analysis of theoretical and simulated results, predicated on an orthogonal test. These results affirm the practicality of simulating quantum heat engines on quantum circuits, offering potential for substantially curtailing the experimental expenses associated with the construction of such engines.
Hong Zeng, Zhao-Qin He, Yun-Ru Fan, Yue Luo, Chen Lyu, Jin-Peng Wu, Yun-Bo Li, Sheng Liu, Dong Wang, De-Chao Zhang, Juan-Juan Zeng, Guang-Wei Deng, You Wang, Hai-Zhi Song, Zhen Wang, Li-Xing You, Kai Guo, Chang-Zheng Sun, Yi Luo, Guang-Can Guo, et al (1) Integrated quantum light source is increasingly desirable in large-scale quantum information processing.~Despite recent remarkable advances, new material platform is constantly being explored for the fully on-chip integration of quantum light generation, active and passive manipulation, and detection. Here, for the first time, we demonstrate a gallium nitride (GaN) microring based quantum light generation in the telecom C-band, which has potential towards the monolithic integration of quantum light source.~In our demonstration, the GaN microring has a free spectral range of 330 GHz and a near-zero anomalous dispersion region of over 100 nm. The generation of energy-time entangled photon pair is demonstrated with a typical raw two-photon interference visibility of 95.5$\pm$6.5%, which is further configured to generate heralded single photon with a typical heralded second-order auto-correlation $g^{(2)}_{H}(0)$ of 0.045$\pm$0.001. Our results pave the way for developing chip-scale quantum photonic circuit.
The three-level system represents the smallest quantum system capable of autonomous cycling in quantum heat engines. This study proposes a method to simulate the steady-state dynamics of a three-level quantum heat engine by designing and implementing superconducting quantum circuits. Following error mitigation, the outcomes from the quantum circuit model designed in this study, when executed on a real quantum device, closely align with theoretical predictions, thereby validating the effectiveness of the circuit model. This study offers a novel approach for investigating three-level quantum heat engines, enabling the verification of theoretical research findings while also reducing the complexity and cost of experimental procedures.
Quantum thermodynamic relationships in emerging nanodevices are significant but often complex to deal with. The application of machine learning in quantum thermodynamics has provided a new perspective. This study employs reinforcement learning to output the optimal cycle of quantum heat engine. Specifically, the soft actor-critic algorithm is adopted to optimize the cycle of three-level coherent quantum heat engine with the aim of maximal average power. The results show that the optimal average output power of the coherent three-level heat engine is 1.28 times greater than the original cycle (steady limit). Meanwhile, the efficiency of the optimal cycle is greater than the Curzon-Ahlborn efficiency as well as reporting by other researchers. Notably, this optimal cycle can be fitted as an Otto-like cycle by applying the Boltzmann function during the compression and expansion processes, which illustrates the effectiveness of the method.
Xueying Zhang, Bin Zhang, Shihai Wei, Hao Li, Jinyu Liao, Cheng Li, Guangwei Deng, You Wang, Haizhi Song, Lixing You, Bo Jing, Feng Chen, Guang-Can Guo, Qiang Zhou Telecom-band integrated quantum memory is an elementary building block for developing quantum networks compatible with fiber communication infrastructures. Towards such a network with large capacity, an integrated multimode photonic quantum memory at telecom band has yet been demonstrated. Here we report a fiber-integrated multimode quantum storage of single photon at telecom band on a laser-written chip. The storage device is a fiber-pigtailed Er3+:LiNbO3 waveguide and allows a storage of up to 330 temporal modes of heralded single photon with 4-GHz-wide bandwidth at 1532 nm and a 167-fold increasing of coincidence detection rate with respect to single mode. Our memory system with all-fiber addressing is performed using telecom-band fiber-integrated and on-chip devices. The results represent an important step for the future quantum networks using integrated photonics devices.
Yun-Ru Fan, Yue Luo, Zi-Chang Zhang, Yun-Bo Li, Sheng Liu, Dong Wang, Dechao Zhang, Guang-Wei Deng, You Wang, Hai-Zhi Song, Zhen Wang, Li-Xing You, Chen-Zhi Yuan, Guang-Can Guo, Qiang Zhou The coexistence of quantum and classical light in the same fiber link is extremely desired in developing quantum communication. It has been implemented for different quantum information tasks, such as classical light coexisting with polarization-entangled photons at telecom O-band, and with quantum signal based quantum key distribution (QKD). In this work, we demonstrate the coexistence of energy-time entanglement based QKD and fiber optical communication at the telecom C-band. The property of noise from the classical channel is characterized with classical light at different wavelengths. With the largest noise, i.e., the worst case, the properties of energy-time entanglement are measured at different fiber optical communication rates. By measuring the two-photon interference of energy-time entanglement, our results show that a visibility of 82.01$\pm$1.10\% is achieved with a bidirectional 20 Gbps fiber optical communication over 40 km. Furthermore, by performing the BBM92 protocol for QKD, a secret key rate of 245 bits per second could be generated with a quantum bit error rate of 8.88\% with the coexisted energy-time entanglement.~Our demonstration paves the way for developing the infrastructure for quantum networks compatible with fiber optical communication.
Jiarui Li, Chenzhi Yuan, Si Shen, Zichang Zhang, Ruiming Zhang, Hao Li, You Wang, Guangwei Deng, Lixing You, Zhen Wang, Haizhi Song, Yunru Fan, Guangcan Guo, Qiang Zhou Discrete frequency-bin entanglement is an essential resource for applications in quantum information processing. In this Letter, we propose and demonstrate a scheme to generate discrete frequency-bin entanglement with a single piece of periodically poled lithium niobate waveguide in a modified Sagnac interferometer. Correlated two-photon states in both directions of the Sagnac interferometer are generated through cascaded second-order optical nonlinear processes. A relative phase difference between the two states is introduced by changing the polarization state of pump light, thus manipulating the two-photon state at the output of the Sagnac interferometer. The generated two-photon state is sent into a fiber polarization splitter, then a pure discrete frequency-bin entangled two-photon state is obtained by setting the pump light. The frequency entanglement property is measured by a spatial quantum beating with a visibility of $96.0 \pm 6.1\%$. The density matrix is further obtained with a fidelity of $98.0 \pm 3.0\%$ to the ideal state. Our demonstration provides a promising method for the generation of pure discrete frequency-bin entanglement at telecom band, which is desired in quantum photonics.
Si Shen, Chenzhi Yuan, Zichang Zhang, Hao Yu, Ruiming Zhang, Chuanrong Yang, Hao Li, Zhen Wang, You Wang, Guangwei Deng, Haizhi Song, Lixing You, Yunru Fan, Guangcan Guo, Qiang Zhou Quantum teleportation can transfer an unknown quantum state between distant quantum nodes, which holds great promise in enabling large-scale quantum networks. To advance the full potential of quantum teleportation, quantum states must be faithfully transferred at a high rate over long distance. Despite recent impressive advances, a high-rate quantum teleportation system across metropolitan fiber networks is extremely desired. Here, we demonstrate a quantum teleportation system which transfers quantum states carried by independent photons at a rate of 7.1$\pm$0.4 Hz over 64-km-long fiber channel. An average single-photon fidelity of $\geqslant$ 90.6$\pm$2.6% is achieved, which exceeds the maximum fidelity of 2/3 in classical regime. Our result marks an important milestone towards quantum networks and opens the door to exploring quantum entanglement based informatic applications for the future quantum internet.
The possibility of utilizing quantum effects to enhance the performance of quantum heat engines has been an active topic of research, but how to enhance the performance by optimizing the engine parameters needs to be further studied. In this study, the temperature difference and dissipation modes affecting the performance of a three-level quantum heat engine were analyzed using an orthogonal test. The results indicated that the dissipation mode dominated the performance of the quantum heat engine. The quantum heat engine performs best when there is only resonance and no detuning; however, when detuning exists, a lower resonance can improve the efficiency by reducing energy losses. Regarding the temperature difference, the best performance was achieved at medium temperature difference owing to the decreasing heat leakage. Finally, the quantum friction caused by the detuning could make the maximal efficiency lower than the Carnot efficiency.
Yun-Ru Fan, Chen Lyu, Chen-Zhi Yuan, Guang-Wei Deng, Zhi-Yuan Zhou, Yong Geng, Hai-Zhi Song, You Wang, Yan-Feng Zhang, Rui-Bo Jin, Heng Zhou, Li-Xing You, Guang-Can Guo, Qiang Zhou Multi-wavelength quantum light sources, especially at telecom band, are extremely desired in quantum information technology. Despite recent impressive advances, such a quantum light source with high quality remains challenging. Here we demonstrate a multi-wavelength quantum light source using a silicon nitride micro-ring with a free spectral range of 200 GHz. The generation of eight pairs of correlated photons is ensured in a wavelength range of 25.6 nm. With device optimization and noise-rejecting filters, our source enables the generation of heralded single-photons - at a rate of 62 kHz with $g^{(2)}_{h}(0)=0.014\pm0.001$, and the generation of energy-time entangled photons - with a visibility of $99.39\pm 0.45\%$ in the Franson interferometer. These results, at room temperature and telecom wavelength, in a CMOS compatible platform, represent an important step towards integrated quantum light devices for the quantum networks.
Shi-Hai Wei, Bo Jing, Xue-Ying Zhang, Jin-Yu Liao, Hao Li, Li-Xing You, Zhen Wang, You Wang, Guang-Wei Deng, Hai-Zhi Song, Daniel Oblak, Guang-Can Guo, Qiang Zhou To advance the full potential of quantum networks one should be able to distribute quantum resources over long distances at appreciable rates. As a consequence, all components in the networks need to have large multimode capacity to manipulate photonic quantum states. Towards this end, a multimode photonic quantum memory, especially one operating at telecom wavelength, remains a key challenge. Here we demonstrate a spectro-temporally multiplexed quantum memory at 1532 nm. Multimode quantum storage of telecom-band heralded single photons is realized by employing the atomic frequency comb protocol in a 10-m-long cryogenically cooled erbium doped silica fibre. The multiplexing encompasses five spectral channels - each 10 GHz wide - and in each of these up to 330 temporal modes, resulting in the simultaneous storage of 1650 modes of single photons. Our demonstrations open doors for high-rate quantum networks, which are essential for future quantum internet.
Shi-Hai Wei, Bo Jing, Xue-Ying Zhang, Jin-Yu Liao, Chen-Zhi Yuan, Bo-Yu Fan, Chen Lyu, Dian-Li Zhou, You Wang, Guang-Wei Deng, Hai-Zhi Song, Daniel Oblak, Guang-Can Guo, Qiang Zhou Quantum networks play an extremely important role in quantum information science, with application to quantum communication, computation, metrology and fundamental tests. One of the key challenges for implementing a quantum network is to distribute entangled flying qubits to spatially separated nodes, at which quantum interfaces or transducers map the entanglement onto stationary qubits. The stationary qubits at the separated nodes constitute quantum memories realized in matter while the flying qubits constitute quantum channels realized in photons. Dedicated efforts around the world for more than twenty years have resulted in both major theoretical and experimental progress towards entangling quantum nodes and ultimately building a global quantum network. Here, we review the development of quantum networks and the experimental progress over the past two decades leading to the current state of the art for generating entanglement of quantum nodes based on various physical systems such as single atoms, cold atomic ensembles, trapped ions, diamonds with Nitrogen-Vacancy centers, solid-state host doped with rare-earth ions, etc. Along the way we discuss the merits and compare the potential of each of these systems towards realizing a quantum network.
Hao Yu, Chenzhi Yuan, Ruiming Zhang, Zichang Zhang, Hao Li, You Wang, Guangwei Deng, Lixing You, Haizhi Song, Zhiming Wang, Guang-Can Guo, Qiang Zhou Heralded single photon source (HSPS) is an important way in generating genuine single photon, having advantages of experimental simplicity and versatility. However, HSPS intrinsically suffers from the trade-off between the heralded single photon rate and the single photon purity. To overcome this, one can apply multiplexing technology in different degrees of freedom to enhance the performance of HSPS. Here, by employing spectral multiplexing and active feed-forward spectral manipulating, we demonstrate a HSPS at 1.5 \mum telecom-band. Our experimental results show that the spectral multiplexing effectively erases the frequency correlation of pair source and significantly improves the heralded single photon rate while keeping the g^(^2^)(0) as low as 0.0006\pm0.0001. The Hong-Ou-Mandel interference between the heralded single photons and photons from an independent weak coherent source indicates a high indistinguishability. Our results pave a way for scalable HSPS by spectral multiplexing towards deterministic single photon emission.
NOON state interference (NOON-SI) is a powerful tool to improve the phase sensing precision, and plays an important role in quantum measurement. In most of the previous NOON-SI experiments, the measurements were performed in time domain where the spectral information of the involved photons was integrated and lost during the measurement. In this work, we experimentally measured the joint spectral intensities (JSIs) at different positions of the interference patterns in both time and frequency domains. It was observed that the JSIs were phase-dependent and show odd (even)-number patterns at $0$ ($\pi$) phase shift; while no interference appeared in time domain measurement, the interference pattern clearly appeared in frequency domain. To our best knowledge, the latter is the first observation of the spectrally resolved NOON state interference, which provides alternative information that cannot be extracted from the time-domain measurement. To explore its potential applications, we considered the interferometric sensing with our setup. From the Fisher information-based analysis, we show that the spectrally resolved NOON-SI has a better performance at non-zero-delay position than its non-spectrally resolved counterpart. The spectrally resolved NOON-SI scheme may be useful for quantum metrology applications such as quantum phase sensing, quantum spectroscopy, and remote synchronization.
Zichang Zhang, Chenzhi Yuan, Si Shen, Hao Yu, Ruiming Zhang, Heqing Wang, Hao Li, You Wang, Guangwei Deng, Zhiming Wang, Lixing You, Zhen Wang, Haizhi Song, Guangcan Guo, Qiang Zhou In this paper, we demonstrate the generation of high-performance entangled photon-pairs in different degrees of freedom from a single piece of fiber pigtailed periodically poled LiNbO$_3$ (PPLN) waveguide. We utilize cascaded second-order nonlinear optical processes, i.e. second-harmonic generation (SHG) and spontaneous parametric down conversion (SPDC), to generate photon-pairs. Previously, the performance of the photon pairs is contaminated by Raman noise photons from the fiber pigtails. Here by integrating the PPLN waveguide with noise rejecting filters, we obtain a coincidence-to-accidental ratio (CAR) higher than 52,600 with photon-pair generation and detection rate of 52.3 kHz and 3.5 kHz, respectively. Energy-time, frequency-bin and time-bin entanglement is prepared by coherently superposing correlated two-photon states in these degrees of freedom, respectively. The energy-time entangled two-photon states achieve the maximum value of CHSH-Bell inequality of S=2.708$\pm$0.024 with a two-photon interference visibility of 95.74$\pm$0.86%. The frequency-bin entangled two-photon states achieve fidelity of 97.56$\pm$1.79% with a spatial quantum beating visibility of 96.85$\pm$2.46%. The time-bin entangled two-photon states achieve the maximum value of CHSH-Bell inequality of S=2.595$\pm$0.037 and quantum tomographic fidelity of 89.07$\pm$4.35%. Our results provide a potential candidate for quantum light source in quantum photonics.
We theoretically investigate the preparation of mid-infrared (MIR) spectrally-uncorrelated biphotons from a spontaneous parametric down-conversion process using doped LN crystals, including MgO doped LN, ZnO doped LN, and In2O3 doped ZnLN with doping ratio from 0 to 7 mol%. The tilt angle of the phase-matching function and the corresponding poling period are calculated under type-II, type-I, and type-0 phase-matching conditions. We also calculate the thermal properties of the doped LN crystals and their performance in Hong-Ou-Mandel interference. It is found that the doping ratio has a substantial impact on the group-velocity-matching (GVM) wavelengths. Especially, the GVM2 wavelength of co-doped InZnLN crystal has a tunable range of 678.7 nm, which is much broader than the tunable range of less than 100 nm achieved by the conventional method of adjusting the temperature. It can be concluded that the doping ratio can be utilized as a degree of freedom to manipulate the biphoton state. The spectrally uncorrelated biphotons can be used to prepare pure single-photon source and entangled photon source, which may have promising applications for quantum-enhanced sensing, imaging, and communications at the MIR range.
We theoretically study the non-equilibrium correlations and entanglement between distant semiconductor qubits in a one-dimensional coupled-mechanical-resonator chain. Each qubit is defined by a double quantum dot (DQD) and embedded in a mechanical resonator. The two qubits can be coupled, correlated and entangled through phonon transfer along the resonator chain. We calculate the non-equilibrium correlations and steady-state entanglement at different phonon-phonon coupling rates, and find a maximal steady entanglement induced by a population inversion. The results suggest that highly tunable correlations and entanglement can be generated by phonon-qubit hybrid system, which will contribute to the development of mesoscopic physics and solid-state quantum computation.
Yun-Ru Fan, Chen-Zhi Yuan, Rui-Ming Zhang, Si Shen, Peng Wu, He-Qing Wang, Hao Li, Guang-Wei Deng, Hai-Zhi Song, Li-Xing You, Zhen Wang, You Wang, Guang-Can Guo, Qiang Zhou With propagating through a dispersive medium, the temporal-spectral profile of laser pulses should be inevitably modified. Although such dispersion effect has been well studied in classical optics, its effect on a single-photon wave-packet, i.e., the matter wave of a single-photon, has not yet been entirely revealed. In this paper, we investigate the effect of dispersion on indistinguishability of single-photon wave-packets through the Hong-Ou-Mandel (HOM) interference. By dispersively manipulating two indistinguishable single-photon wave-packets before interfering with each other, we observe that the difference of the second-order dispersion between two optical paths of the HOM interferometer can be mapped to the interference curve, indicating that (1) with the same amount of dispersion effect in both paths, the HOM interference curve must be only determined by the intrinsic indistinguishability between the wave-packets, i.e., dispersion cancellation due to the indistinguishability between Feynman paths; (2) unbalanced dispersion effect in two paths cannot be cancelled and will broaden the interference curve thus providing a way to measure the second-order dispersion coefficient. Our results suggest a more comprehensive understanding of the single-photon wave-packet and pave ways to explore further applications of the HOM interference.
We experimentally demonstrate a real-time quantum random number generator by using a room-temperature single-photon emitter from the defect in a commercial gallium nitride wafer. Thanks to the brightness of our single photon emitter, the raw bit generation rate is ~1.8 MHz, and the unbiased bit generation rate is ~420 kHz after von Neumann's randomness extraction procedure. Our results show that commercial gallium nitride wafer has great potential for the development of integrated high-speed quantum random number generator devices.
Spectrally intrinsically uncorrelated biphoton states generated from nonlinear crystals are very important but rare resources for quantum photonics and quantum information applications. Previously, such biphoton states were generated from several kinds of crystals, however, their wavelength ranges and nonlinear efficiencies were still limited for various applications. In order to explore new crystal for wider wavelength range and higher nonlinear efficiency, here we theoretically study the generation of spectrally uncorrelated biphoton states from 14 crystals in the `BBO family', including BBO, CLBO, KABO, KBBF, RBBF, CBBF, BABF, BiBO, LBO, CBO, LRB4, LCB, YCOB, and GdCOB. They satisfy three kinds of group-velocity matching condition from near-infrared to telecom wavelengths. Furthermore, heralded single photons can be generated with a purity as high as 0.98, which is achieved without any narrow filtering. The indistinguishability of photons from independent sources is examined by the Hong-Ou-Mandel interference, which results in a visibility of 98% also without any further filtering, i.e., photons from different heralded single-photon sources are highly indistinguishable. Our study may provide single-photon sources with good performance for quantum information processing at near-infrared and telecom wavelengths.
Bin Li, Ruiming Zhang, Yong Wang, Hao Li, Lixing You, Zhonghua Ou, Heng Zhou, Yun Ling, Yunxiang Wang, Guangwei Deng, You Wang, Haizhi Song, Kun Qiu, Qiang Zhou Photon counting optical time-domain reflectometry (PC-OTDR) based on the single photon detection is an effective scheme to attain the high spatial resolution for optical fiber fault monitoring. Currently, due to the spatial resolution of PC-OTDR is proportional to the pulse width of a laser beam, short laser pulses are essential for the high spatial resolution. However, short laser pulses have a large bandwidth, which would be widened by the dispersion of fiber, thereby causing inevitable deterioration in spatial resolution, especially for long-haul fiber links. In this letter, we propose a scheme of dispersion independent PC-OTDR based on an infinite backscatter technique. Our experimental results -with more than 50 km long fiber - show that the spatial resolution of the PC-OTDR system is independent with the total dispersion of the fiber under test. Our method provides an avenue for developing the long-haul PC-OTDR with high performance.
Bin Li, Guangwei Deng, Ruiming Zhang, Zhonghua Ou, Heng Zhou, Yun Ling, Yunxiang Wang, You Wang, Kun Qiu, Haizhi Song, Qiang Zhou Single photon detector (SPD) has a maximum count rate due to its dead time, which results in that the dynamic range of photon counting optical time-domain reflectometry (PC-OTDR) de-creases with the length of monitored fiber. To further improve the dynamic range of PC-OTDR, we propose and demonstrate an externally time-gated scheme. The externally time-gated scheme is realized by using a high-speed optical switch, i.e. a Mach-Zehnder interferometer, to modulate the back-propagation optical signal, and to allow that only a certain segment of the fiber is monitored by the SPD. The feasibility of proposed scheme is first examined with theoretical analysis and simulation; then we experimentally demonstrate it with our experimental PC-OTDR testbed operating at 800 nm wavelength band. In our studies, a dynamic range of 30.0 dB is achieved in a 70 meters long PC-OTDR system with 50 ns external gates, corresponding to an improvement of 11.0 dB in dynamic range comparing with no gating operation. Furthermore, with the improved dynamic range, a successful identification of a 0.37 dB loss event is detected with 30-seconds accumulation, which could not be identified without gating operation. Our scheme paves an avenue for developing PC-OTDR systems with high dynamic range.
The Casimir force between two parallel plates separated by anisotropic media is investigated. We theoretically calculate the Casimir force between two parallel plates when the interspace between the plates is filled with anisotropic media. Our result shows that the anisotropy of the material between the plates can significantly affect the Casimir force, especially the direction of the force. If ignoring the anisotropy of the in-between material makes the force to be repulsive (attractive), by contrast, taking the anisotropy into account may produce an extra attractive (repulsive) force. The physical explanation for this phenomenon is also discussed.
Miao-Lei Zhang, Guang-Wei Deng, Shu-Xiao Li, Hai-Ou Li, Gang Cao, Tao Tu, Ming Xiao, Guang-Can Guo, Hong-Wen Jiang, Irfan Siddiqi, Guo-Ping Guo We have designed and fabricated a half-wavelength reflection line resonator (RLR) that consists of a pair of two coupled microstrip lines on a GaAs/AlGaAs heterostructure. By changing the top gate voltage on a square of two dimensional electron gas under the resonator, a large range of the quality factors can be obtained. Energy loss in the two-dimensional electron gas can be minimized, thus realizing a versatile resonator suitable for integration with semiconductor quantum circuits.
The novel concept of non-compensated n-p codoping has made it possible to create tunable intermediate bands in the intrinsic band gap of TiO2, making the codoped TiO2 a promising material for developing intermediate band solar cells (IBSCs). Here we investigate the quantum efficiency of such IBSCs within two scenarios - with and without current extracted from the extended intermediate band. Using the ideal equivalent circuit model, we find that the maximum efficiency of 57% in the first scenario and 53% in the second are both much higher than the Shockley-Queisser limit from single gap solar cells. We also obtain various key quantities of the circuits, a useful step in realistic development of TiO2 based solar cells invoking device integration. These equivalent circuit results are also compared with the efficiencies obtained directly from consideration of electron transition between the energy bands, and both approaches reveal the intriguing existence of double peaks in the maximum quantum efficiency as a function of the relative location of IBs.