Tuning coupling between superconducting resonators with collective qubits

Tuning coupling between superconducting resonators with collective qubits

Qi-Ming Chen Q.M.Chen is currently at the Department of Chemistry, Princeton University as a visiting scholar. Department of Automation, Tsinghua University, Beijing, 100084, China Center for Quantum information Science and Technology, Tsinghua National Laboratory for Information Science and Technology, Beijing 100084, China    Re-Bing Wu rbwu@tsinghua.edu.cn Department of Automation, Tsinghua University, Beijing, 100084, China Center for Quantum information Science and Technology, Tsinghua National Laboratory for Information Science and Technology, Beijing 100084, China    Luyan Sun Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China    Yu-xi Liu Center for Quantum information Science and Technology, Tsinghua National Laboratory for Information Science and Technology, Beijing 100084, China Institute of Microelectronics, Tsinghua University, Beijing 100084, China
July 21, 2019
Abstract

By coupling multiple artificial atoms simultaneously to two superconducting resonators, we construct a quantum switch that controls the resonator-resonator coupling strength from zero to a large value proportional to the number of qubits. This process is implemented by switching the qubits among different subradiant states, where the microwave photons decayed from different qubits interfere destructively so that the coupling strength keeps stable against environmental noise. Based on a two-step control scheme, the coupling strength can be switched at the nanosecond scale while the qubits are maintained at the coherent optimal point. We also use the quantum switch to connect multiple resonators with a programmable network topology, and demonstrate its potential applications in quantum simulation and scalable quantum information storage and processing.

I Introduction

Superconducting circuit is of increasing importance in simulating various quantum physics Georgescu et al. (2014) as well as in quantum information processing Schoelkopf and Girvin (2008); Gu et al. (2017). As an elementary component, transmission line resonators play a central role in microwave photon storage and transmission Blais et al. (2004); Wallraff et al. (2004). Architectures with two Sun et al. (2006); Mariantoni et al. (2008); Reuther et al. (2010); Baust et al. (2015); Helmer et al. (2009); Johnson et al. (2010), three Wang et al. (2011); Mariantoni et al. (2011), and even larger arrays of superconducting resonators Raussendorf and Harrington (2007); DiVincenzo (2009); Steffen et al. (2011); Koch et al. (2010); Underwood et al. (2012) have been proposed and demonstrated in recent experiments. Current nano-fabricating technologies provide superconducting circuits with more possibilities in the study of large quantum systems Fink et al. (2009); Filipp et al. (2011); van Loo et al. (2013); Mlynek et al. (2014); Lambert et al. (2016).

Processing information in superconducting circuits requires tunable coupling between different components, i.e., the quantum switch. In the literature, various quantum switches have been investigated between two qubits Blais et al. (2003); Berkley et al. (2003); McDermott et al. (2005); xi Liu et al. (2006); Majer et al. (2007); Bialczak et al. (2011), one qubit and one resonator Cleland and Geller (2004); Sillanpaa et al. (2007); Leek et al. (2010); Eichler et al. (2012) or transmission line Yin et al. (2013), and two resonators Sun et al. (2006); Mariantoni et al. (2008); Reuther et al. (2010); Baust et al. (2015). Typically, the resonator-resonator (r-r) coupling switch is realized by coupling one artificial atom simultaneously to two superconducting resonators Sun et al. (2006); Mariantoni et al. (2008); Reuther et al. (2010); Baust et al. (2015). By tuning the transition frequency or flipping the state population of the qubit, r-r coupling can be adjusted in a certain range. However, the r-r coupling is limited by the dispersive coupling strength between the qubit and the resonators, and it is unstable because that the qubit cannot always stay at the coherent optimal point. In this paper, we find that the coupling strength can be extended into a much wider and more stable range by adding more qubits that are prepared at certain collective states.

The idea behind can be dated back to 1950s, when Dicke found that collective two-level atoms exhibit many new features that cannot be seen in a single atom or an ensemble of non-collective atoms Dicke (1954). One of the most fascinating phenomena is that decoherence of the atoms can be effectively suppressed if they face the same environment and are engineered at the so called subradiant states Freedhoff and Kranendonk (1967); Stroud et al. (1972); Gross and Haroche (1982); Pavolini et al. (1985); Fink et al. (2009); Filipp et al. (2011). The quantum switch proposed in this paper couples collective two-level artificial atoms simultaneously to two superconducting resonators, in which the r-r coupling strength can be effectively tuned by engineering the subradiant states of the collective qubits while the decoherence can be effectively suppressed.

The rest of the paper is organized as follows. In Sec. II and III, we introduce how to use collective qubits to realize a strong and stable coupling between two superconducting resonators, as well as how to control the coupling strength within nanoseconds. Then in Sec. IV, we use the quantum switch to connect multiple superconducting resonators with a programmable network topology, and show its possible applications in quantum simulation and quantum information storage and processing. Finally, we draw conclusions and present further discussions in Sec. V.

Ii The quantum switch with collective superconducting qubits

Figure 1: (Color online) Circuit of the quantum switch. Two superconducting resonators (blue) and (yellow) are coupled through two transmon qubits (green). Transition frequency of an individual qubit can be tuned by the local superconducting coils, and direct qubit transitions can be driven by the local microwave drive lines. Coupling strength between the qubits and resonators can be adjusted by optimizing the relative location between the two components.

Consider the system shown in FIG. 1 where two superconducting resonators are mediated by two-level artificial atoms, the Hamiltonian reads

where direct coupling between different qubits have been fairly omitted; () and () are the annihilation (creation) operators of the two resonators; () is the lowing (raising) operator of the th qubit in the collection, and is the angular momentum operator along the -axis. To simplify the system, we suppose that all the N qubits have the same frequency , and they are equally coupled to the two resonators and with the same coupling strengths and . This assumption can be realized by optimizing the locations of the individual superconducting qubits with respect to the resonators Delanty et al. (2011); van Loo et al. (2013). In these regards, operators for the qubits can be simplified by using the following collective operators

(2)

Suppose that all the qubits are largely detuned from the two resonators, we introduce the following transformation in order to manifest the effective interaction between the two resonators

(3)

where for . Using the Baker-Hausdorff Lemma Sakurai (1994) and omitting small terms above the second order, the dispersive Hamiltonian reads

(4)

where for , and .

According to the commutation relation

(5)

the collective operators defined in Eq. (2) can be simply treated as angular momentum Sakurai (1994), and are thus called the collective angular momentum operators. Eigenstates of the operator are those

(6)
(7)

where or is related to the total angular momentum, and is its projection on the axis. Thus, according to the last term of Eq. (4) the effective coupling strength between the two resonators can be varied from zero to when is even, or from to when is odd. In principle, the maximum coupling strength can be arbitrarily strong when the qubit number is very large, so that interaction between the superconducting resonators can be tuned in a rather wide range by engineering the state of the collective qubits.

Iii Control of the resonator-resonator coupling strength

iii.1 Stable coupling against environmental noise

Suppose and are the decay rate of the resonators and , and are the energy and phase decay rate of a single qubit, the master equation for the system reads

(8)

where is the Lindblad superoperator. We have assumed in the above equation that the qubits interact with the same environment. This requirement has been experimentally proven in Refs. Fink et al. (2009); Filipp et al. (2011), though it cannot be fully satisfied owing to the currently imperfect nano-fabricating technology. The worst situation with totally different environments for individual qubits will be numerically simulated in Sec. IV.

To avoid the energy and phase relaxation of the collective qubits, we consider the eigenstates of both the operators and , which are the subradiant states of the qubit collection

(9)

where is an abbreviation of the state with . On one hand, different subradiant states correspond to different angular momentum of , so that they can be used to control the r-r coupling strength. On the other hand, the subradiant states are immune to the energy and phase relaxation terms described by the superoperators and Scully (2015), so that these states can last for a very long time in an open environment. Consider also that the individual qubits are always stayed at the coherent optimal point, the quantum switch we designed provides a rather stable coupling strength between two superconducting resonators, which is not influenced by the environmental noise.

iii.2 Two-step control scheme with nanoseconds coupling tunning

To engineer a stable r-r coupling with tunable strengths, we need to switch the collective qubits among subradiant states with different collective angular momenta. Suppose that the qubits are initially prepared at the subradiant state which corresponds to an effective r-r coupling strength , our aim is to switch the coupling strength into as fast as possible. Consider that engineering of the subradiant states requires the multi-qubit interactions which are usually very small in the dispersive regime, we propose a two-step control scheme that engineers the r-r coupling strength and the subradiant state seperately.

Consider a special set of subradiant states in which all the qubits are grouped into pairs. To simplify notation, we define the subradiant state formed by qubits with collective angular momentum as follows

(10)

where is the singlet state composed of a two-qubit pair. Thus, engineering of the subradiant states can be simplified by engineering every two qubits in the collection Scully (2015).

Our first step is to use the local microwave drive lines to flip one qubit in the first pairs from the ground state into excited state, i.e., we engineer pairs from to . Under typical parameter set in superconducting circuits, this process transfers the collective angular momentum into at the rate of GHz, which equivalently engineers the r-r coupling strength into at the nanosecond scale.

To suppress the environmental noise, the second step is to engineer the qubits into the subradiant state while keeping the coupling strength unchanged. According to the commutation relation

(11)

entanglement in the qubit collection can be generated by using the virtual qubit-qubit interaction, i.e., the fifth term of Eq. (4), while the collective angular momentum of the qubits can be maintained throughout the process. In particular, this virtual interaction couples the qubits with each other according to the Schrödinger equation

(12)

where is a normalization factor, . After a time duration , the qubits will evolute to the following entangled state

(13)

Then, we apply a single-qubit phase gate on the first qubits in order to correct the relative phase from into . This process can be realized by a combination of and gates with local microwave drive lines, or by introducing off-resonant drives that induce ac-Stark shift on the qubits Blais et al. (2007). In the dispersive regime where for , the second step can be implemented within microseconds. To sum up, the two-step control scheme realizes a nanosecond-scale switch for controlling the r-r coupling strength, followed with a microsecond-scale step to stabilize coupling strength.

It is also possible to apply other control methods such as reservoir engineering Poyatos et al. (1996); Geerlings et al. (2013); Leghtas et al. (2013); Shankar et al. (2013), FLICFORQ Rigetti et al. (2005), and sideband transitions Leek et al. (2009) to switch the qubits among different subradiant states. These methods, however, control the coupling strength at the microsecond scale which is much slower than our method, and they may cause unwanted changes in the coupled superconducting resonators. One may also resort to optimal Khaneja et al. (2005); Heeres et al. (2017) or time-optimal control methods Chen et al. (2015) to find feasible control pulses for the system, but the computational complexity should be considered as a limitation which will not be discussed here. More detailed discussions of the control methods in superconducting circuits can be found in Ref. Blais et al. (2007), and it is still an open question on engineering of subradiant states in this system.

Iv Superconducting resonator network with programmable interaction

Figure 2: (Color online) Schematic of the programmable interactions in a superconducting resonator network. The resonators are connected with each other by quantum switches, so that the interaction among different resonators can be effectively controlled by engineering the quantum switches mediated in between. The dashed rectangle shows a controlled three-resonator chain which is numerically simulated in Figure 2.

Based on the quantum switch proposed, programmable coupling among multiple superconducting resonators can be realized in the architecture illustrated in Fig. 2. Qualitatively speaking, interaction between any two resonators can be controlled by the quantum switches mediated in between. Consider for simplicity a three-resonator chain as represented in the dashed rectangle, the system’s Hamiltonian reads

where () for are the creation (annihilation) operators for the resonators A (red), B (yellow), and C (blue); (orange) and (green) are the collective angular momentum operators for the two qubit collections and . Suppose that the qubits are largely detuned from the three resonators, interaction between different resonators can thus be either direct or virtual according to the resonant frequencies which correspond to the following two types of applications.

iv.1 Programmable quantum simulator

Suppose that frequencies of the three resonators A, B, and C hare identical, the system can be well described by the following dispersive Hamiltonian

(15)

where , other parameters are defined in the same way as in the two-resonator case. Suppose that the qubits are prepared at the subradiant state, the last three terms of the Hamiltonian can be eliminated and we obtain the 1D Su-Schrieffer-Heeger (SSH) interaction in the system, i.e.,

(16)

where is the resonator number in the chain (in this case ). As demonstrated in Refs. Koch et al. (2010); Underwood et al. (2012), the same architecture without quantum switches has been successfully fabricated in recent experiments, and this superconducting resonator network is proven to be very useful in simulating quantum phase transitions in condensed matter physics. Different from that, the effective r-r couplings in our system can be switched on and off by controlling the state of the superconducting qubits, so that the network topology of the quantum simulator is programmable after circuit fabrication. Moreover, the coupling strength between two adjacent resonators can be adjusted from zero to (suppose is the qubit number in the th quantum switch). This results in a quantum simulator with programmable interactions between different components, which can be used to implement various quantum simulations, such as quantum phase transition and many-body physics, in only one superconducting circuit.

iv.2 Quantum information storage and processing

Suppose that A and C are high-quality storage resonators with the same frequency , which is far detuned from the low-quality bus resonator B with frequency , we apply the following transformation on Eq. (15) to manifest the effective interaction between the two storage resonators

(17)

where . Using the Baker-Hausdorff Lemma and omitting the higher-order terms, we obtain the following Hamiltonian

(18)

where we have assumed that the qubits are prepared at the subradiant states. In this system, the interaction between the two storage resonators is described by the last term, which is proportional to the product of the collective angular momentum of different qubit collections. For the resonator case with only the st and the th resonators being the storage resonators, the effective interaction between the two resonators reads

(19)

where , is the resonator number in the chain. For simplicity, we suppose that every qubit collection consists superconducting qubits, the r-r coupling strength between the two distant storage resonators then varies from zero to . Even if is extremely small in the dispersive regime, the r-r coupling strength can still be relatively large in principle as long as . In this regard, this architecture enables quantum information storage in the individual storage resonators when the corresponding quantum switches are turned off, as well as quantum information processing between distant storage resonators when the switches are turned on.

Figure 3: (Color online) Master equation simulation of the three-resonator chain presented in the dashed rectangle in Figure 2. The two storage resonators A (red) and C (blue) exchange microwave photons with each other when both of the quantum switches are turned on, while state of the bus resonator B (yellow) is not influenced over time (upper figure). This interaction can be effectively turned off by switching off one of the quantum switches (lower figure), where the red dashed line shows the ideal decay process of a bare storage resonator A. Parameters of the system are listed as follows: GHz, GHz, GHz, MHz, MHz, KHz, KHz, MHz 222Numerical simulation is implemented by using the QuTip toolbox in Python, where the photon number is truncated at for the two storage resonators and for the bus resonator Johansson et al. (2013)..

Figure 2 simulates the three resonator case where the two quantum switches are both turned on, or one of the switch is turned off. The initial state of the three resonators are engineered at , , and , and the environments for different qubits are supposed to be different which corresponds to the worst case one can observe in the experiments. As expected, the exchange of microwave photons between the two storage resonators can be effectively controlled by the quantum switch, i.e., by engineering the subradiant states of the qubit collection. When the two switches are turned on, the effective coupling strength between the two storage resonators achieves MHz in the typical parameter set. However, Rabi flopping between the resonators decays at the ns time scale induced by the low-quality bus resonator, which shows a trade-off between increasing the r-r coupling strength and decreasing the parameter . Because that the r-r coupling is proportional to the qubit number, this relaxation can be effectively suppressed by increasing the frequency detuning while the coupling strength is maintained by increasing the qubit number .

When one of the two switches is turned off, the storage resonators cannot interact with each other and the photons are stored in the individual resonators. Due to the relaxation of both the storage resonators and the qubit collections, state of the storage resonator A decays slowly with time. Compared with the decoherence of the bare storage resonator (dashed red lines), the quantum switch affects the photon storage only by a small ratio so that the the quantum information is protected in the individual resonators. Combined with the state preparation Law and Eberly (1996); xi Liu et al. (2004); Hofheinz et al. (2008, 2009) and non-demolition measurement schemes Wallraff et al. (2005); Schuster et al. (2007); Johnson et al. (2010) in literature, this results in a scalable quantum information processor with full control of quantum information read-in, storage, processing, and read-out, which will be studied in our future researches.

V Conclusions and discussions

In conclusion, we design a quantum switch that controls the coupling strength between two superconducting resonators in a rather wide range. This switch consists a collection of superconducting qubits which are coupled simultaneously to two resonators. By switching the collective qubits among different subradiant states, the coupling strength varies from zero to a large value proportional to the number of qubits. Moreover, the quantum switch is proved to be very stable in regards of energy and phase relaxation in an open environment, and the switching of the coupling strength is implemented within nanoseconds by using a two-step control method. In addition, we use the quantum switch to connect multiple resonators in a quantum network, in which the coupling strength between any two components can be adjusted by controlling the quantum switch. According to the different cases where the resonators are on resonant or largely detuned from each other, this resonator network can be used to simulate quantum phase transition and many-body physics in only one superconducting circuit, or store and process quantum information in a scalable manner.

Although transmon qubits are addressed in this paper, other types of superconducting qubits such as charge, flux, and phase qubits can also be used with little modification. Natural qubits such as neutral atoms, ions, and spins should also be available in principle. This type of hybrid quantum system would benefit from the long coherence time of the natural qubits as well as the uniform coupling between the resonators and the qubits, and provides a more reliable quantum switch than that designed in this paper. However, this proposal may be limited by the weak coupling strength between the natural qubits and the superconducting resonator Xiang et al. (2013), and these problems will be studied in the future.

Acknowledgement

R.B.W acknowledges the support of the National Natural Science Foundation of China (Grant No. 61134008, 61374091, and 61773232) and National Key Research and Development Program of China (Grant No. 2017YFA0304300). LS acknowledges the support of the National Natural Science Foundation of China Grant No.11474177 and 1000 Youth Fellowship program in China. Y.X.L. acknowledges the support of the National Basic Research Program of China Grant No. 2014CB921401 and the National Natural Science Foundation of China Grant No. 91321208.

References

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