Quantum state conversion in opto-electro-mechanical systems via shortcut to adiabaticity
Adiabatic process has found many important applications in modern physics, the distinct merit of which is that it does not need accurate control over the timing of the process. However, it is a slow process, which limits the application in quantum computation, due to the limited coherent times of typical quantum systems. Here, we propose a scheme to implement quantum state conversion in opto-electro-mechanical systems via shortcut to adiabaticity, where the process can be greatly speeded up while the precise timing control is still not necessary. In our scheme, only by modifying the coupling strength, we can achieve fast quantum state conversion with high fidelity, where the adiabatic condition does not need to be met. In addition, the population of the unwanted intermediate state can be further suppressed. Therefore, our protocol presents an important step towards practical state conversion between optical and microwave photons, and thus may find many important applications in hybrid quantum information processing.
Hybrid quantum systems hybrid () may consolidate the advantages of different systems, and thus may find many important applications in quantum information processing. Recently, opto-electro-mechanical systems oem1 (); oem2 () that interfacing optical and microwave photons have attracted considerable attention due to the advanced fabrication of superconducting circuits that support microwave photons and the scalable integrated optical photonic circuit techniques. Through opto-electro-mechanical systems, one can efficient up-conversion microwave information to the optical counterpart, and thus enable the transmission of the information through optical fibres in a low-loss way. Therefore, great efforts have been paid to the conversion between microwave and optical fields Wang (); tian (); science (); Hill (); tianlin (); Lukin (); NJP (); xue1 (); xue2 (); xue3 (); wang2 (). However, the conversion process usually uses the adiabatic passage, which requires a long operation time to satisfy the adiabatic criteria, and thus decoherence may induce unacceptable loss.
One possible way out of the difficulty is the so-called ”superadiabatic transitionless driving (SATD)” Berry (); TQD1 (); TQD2 () or ”shortcut to adiabatic (STA)” chenxi (); PRL2012 (); PRA2016 (); oe2016 (); PRA2015 () protocol, where the conversion process is speeded up and still keep the merits of the adiabatic passage. In this protocol, a system will force to follow exactly the instantaneous eigenstates of its Hamiltonian by applying additional a precisely controlled field to cancel nonadiabatic transitions between the instantaneous eigenstates Chen2012 (); iterative (); Chen2014 (); s1 (); s2 (); s3 (); song (); s4 (); Deng (). In particular, it is found that the protocol can still be simplified du (); Y.C.Li (); dress (); xiayan (); 15 (); zhangshou (); 16 (); liubaojie (), only by modifying the driving fields of the adiabatic case. Moreover, it is also indicated that the populations of the unwanted intermediate state can be suppressed by properly choosing the control parameters, thus reduces its influence and leads to higher fidelity dress (); 17 ().
Here, we propose a scheme to achieve quantum state conversion (QSC) between microwave and optical modes in opto-electro-mechanical systems via shortcut to adiabaticity. Our scheme holds the advantages of the QSC in an adiabatic way but does not require the adiabatic condition to be met, and thus has potential applications in hybrid quantum information processing.
Ii The system and its adiabatic dynamics
We consider an opto-electro-mechanical system, as illustrated in Fig.1(a), where a mechanical resonator simultaneously coupled to an optical cavity and a microwave cavity via dispersive coupling and each cavity modes are under external driving with frequencies (=1, 2) optomechanics (); optomechanics2 (). Follow the standard linearization procedure optomechanics (), the system can be described by
where we have assumed (here and hereafter); , and are the annihilation operators for the optical, microwave and mechanical modes; (=1, 2) is the frequency of the cavity mode, is the mechanical oscillator’s frequency, and is the detuning between cavity mode and external driving; (=1, 2) is the effective linear coupling that is proportional to the driving amplitude applied to cavity , which are tunable by varying the driving field tunable coupling () with and are the effective single-photon coupling strength and photon number inside the cavity, respectively. Note that the Hamiltonian in Eq. 1 is written in a displaced frame, which has a form readily for QSC, and thus the quantum state to be transferred here sits atop a classical coherent state with large number of photons Wang ().
We consider that the case of strong coupling with external driving in the first red sideband of the mechanical mode, i.e., . Moving to the interaction picture, and using the rotating wave approximation, we obtain that
where we have assumed the basis vectors as , , and , corresponding to states with one excitation of the optical cavity, the mechanical oscillator and the microwave cavity, respectively. Noted that the state transfer protocol is confined in single excitation subspace formed by in low temperatures. Three instantaneous eigenvectors of the Hamiltonian in Eq. (2) can be described by
and the corresponding three eigenvalues are , , and , where and , as shown in Fig.1(c). In this system, and are the ”bright” mode, which are superpositions of the cavity , cavity and the mechanical mode ; while is the ”dark” mode, which decouples from the mechanical mode due to destructive interference. If one initially prepared a excitation in the microwave mode , through the ”dark” state passage, one can adiabatically convert the excitation to the optical mode , and vice versa science ().
Iii Shortcut to adiabatic quantum state conversion
iii.1 The protocol
We now consider the case of speeding up the adiabatic QSC process, where the adiabatic condition is not met. In the adiabatic basis , the Hamiltonian in Eq. (2) becomes
, , and are the pauli matrix for spin 1 system, which obey the commutation relation with being the Levi-Civita symbol. The second term of the Hamiltonian in Eq. (III.1) corresponds to nonadiabatic transitions among the dark state and the two bright states when the adiabatic condition is not met well, as shown in Fig. 1 (c).
In order to correct the nonadiabatic leakage, a correction Hamiltonian
is introduced, so that the total Hamiltonian becomes
for transitionless quantum driving. Therefore, in the adiabatic basis, the total modified Hamiltonian becomes
which does not have the nonadiabatic transitions. However, the Hamiltonian in Eq. (6) refers to the direct coupling between the microwave and optical modes, which is hard to be directly induced experimentally.
To overcome this obstacle, we look for another correction Hamiltonian via the dressed state method dress (), so that we can speed up the QSC by only modifying the pulse shape of the coupling strength in Eq. (2). Here, we take
so that the total Hamiltonian in the adiabatic basis now becomes
where the modified couplings are
From the discussion in the last section, we know that this can not be achieved if we use the adiabatic eigenstates as the conversion channel. However, it only requires the initial and the final states are in the adiabatic eigenstates. Therefore, we move to the dressed state picture with respect to . After these two transformations, we find that the total Hamiltonian becomes
As the modified Hamiltonian should be designed to cancel out the unwanted off-diagonal elements, the controlled parameters should be
where, for a simplest nontrivial example, we may set .
iii.2 Numerical simulations
For the time dependence coupling, we choose , . In order to obtain such coupling in opto-electro-mechanical systems, modulating the external driving field is a feasible way, as with being the photon number inside a cavity. In the strongly-driven condition, external driving field excites large number of photons, , in the cavity photons (). Therefore, the amplitude and phase of the couplings can be adjusted in a broad range tunable coupling (). Note that during the superadiabatic correction process, one should guarantee that each corrected couplings cannot exceed its original couplings’ peak amplitude , i.e., we need to ensure that . This constraint implies that we can only speed up the process with an minimal time , through our numerical verification. As shown in Fig. 2(a), the final fidelity approximately reaches 98.4% and 99.9% for and without decoherence, respectively. Therefore, we set as the adiabatic case for our reference.
We now compare the performance of the adiabatic and superadiabatic QSC under dissipation. The performance of the QSC is evaluated by considering the influence of dissipation using the Markovian master equation
where is the density matrix of the considered system. is the Lindblad superoperator with ; and are the decay rate of the optical cavity and microwave cavity due to the loss of photons inside cavity; is the decay rates of the mechanical oscillator . Here, we take the decoherence induced by the mechanics mode for having thermal excitation at low temperatures. Here, we choose MHz and the decay rates , has already been demonstrated experimentally photons (), where a fidelity of can be obtained. As shown in Fig. 2(b), when , we can only obtain a fidelity of 73.80% for the QSC. Correspondingly, when , one finds that the conversion fidelity reaches 96.53% as shown in Fig. 2(c).
iii.3 Suppression of the intermediate state population
In most dissipation system, operation time and decoherence are two major factor influencing the final fidelity. There is a trade-off between operation time and decoherence wang2 (). When the operation time is long enough to satisfy the adiabatic condition well, high fidelity can be obtained, while dissipation will destroy it due to long time integral. When the operation time is too short, non-adiabatic leakage may lead to poor performance during the conversion procedure. Through the superadiabatic correction, there is no need to worry about this trade-off, for we release the adiabatic condition. The decoherence property of the system becomes our major concern.
However, as shown in Fig. 2(c), the population of the intermediate mechanical mode is pretty large, which is what we should try to avoid. The intermediate state may decay to the ground state, and thus reduce the conversion fidelity. Therefore, if the decay of the intermediate mode is large, one of the main issue of QSC mediated by a quantum bus is to find ways of reducing the population of intermediate state. The population of the intermediate level is determined by , i.e., we may try to reduce in order to suppress the population. Therefore, we generalize Eq. (13) to
by introducing an auxiliary function , with the squeeze parameter been optimized for each operation time. Note that the pulse shape is different from that of in Ref. dress (), and thus the auxiliary function is different.
To illustrate the suppression of the intermediate state population, an auxiliary function with for operation time is chosen as an example, as shown in Fig. 3(a), a fidelity of 96.45% can now be obtained and the maximum population of the intermediate state drops from 0.63 (), as shown in Fig. 2(c), to 0.34 (). While the largest we can get here is 0.85, for we need to guarantee that the peak value of and is no larger than the peak value of original strength and , respectively. We also choose a set of range from 0.85 to 0, the population of the intermediate state witness a significant decrease as we increase , as shown in Fig. 3(b), which shows our way of reducing the population of the mechanical mode is quite effective. The pulse shapes for is plotted in Fig. 3(c). Under this suppression, we note that the fidelity is deceased instead of increased. This is because the decay of the optical cavity is the main decoherence source in our system, the suppression requires further modification of the pulse shape, which deviates from the optimal one and thus results in slight decrease of the final state fidelity. If the decay rate of the intermediate state is larger, the suppression of the immediate population will be more important, and the suppression will lead to the increase of the final fidelity, as shown in Fig. 3(d).
We note that when the operation time is longer, the population of the immediate mechanical mode can also be suppressed. Therefore, we further explore the conversion fidelity for a slower process. For , the fidelity is 92.86% in Fig. 4(a) and the corresponding superadiabatic pulse shapes have been plotted in Fig. 4(b). Meanwhile, by introducing another auxiliary function with , the fidelity we can obtain is 92.81%, as shown in Fig. 4(c). It is obvious that the fidelity is also slightly decreased as we expained in the above. Meanwhile, the fidelity is smaller that the case of . This is quite natural as the decay rate of the mechanical mode is small, and thus the operation time here is the most important decoherence source. The largest we can choose for is 0.69, since we still need to guarantee that peak amplitude of is no larger than the peak amplitude of . Hence, we also choose a set of A range from 0.69 to 0, the maximum population of the intermediate state also witness a significant decrease, as shown in Fig. 4(d).
In conclusion, we have proposed a scheme to realize the superadiabatic QSC process in opto-electro-mechanical system, which can significantly speed up the adiabatic procedure by using dressed state. Our scheme possess the following remarkable advantages. Firstly, there is no direct coupling between the target and initial modes in the Hamiltonian, and thus is feasible experimentally. Secondly, during the whole evolution, the adiabatic condition is released, and thus fast and high fidelity can still be achieved compared to the conventional ones. Therefore, our protocol presents an important step towards practical state conversion between optical and microwave photons, and thus may find many applications in hybrid quantum information processing.
Acknowledgements.This work was supported in part by the NFRPC (No. 2013CB921804), the NKRDPC (No. 2016YFA0301803), and the NSF of Jiangsu province (No. BK20140588).
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