Circuit QED: Cross-Kerr-effect induced by a superconducting qutrit without classical pulses
The realization of cross-Kerr nonlinearity is an important task for many applications in quantum information processing. In this work, we propose a method for realizing cross-Kerr nonlinearity interaction between two superconducting coplanar waveguide resonators coupled by a three-level superconducting flux qutrit (coupler). By employing the qutrit-resonator dispersive interaction, we derive an effective Hamiltonian involving two-photon number operators and a coupler operator. This Hamiltonian can be used to describe a cross-Kerr nonlinearity interaction between two resonators when the coupler is in the ground state. Because the coupler is unexcited during the entire process, the effect of coupler decoherence can be greatly minimized. More importantly, compared with the previous proposals, our proposal does not require classical pulses. Furthermore, due to use of only a three-level qutrit, the experimental setup is much simplified when compared with previous proposals requiring a four-level artificial atomic systems. Based on our Hamiltonian, we implement a two-resonator qubits controlled-phase gate and generate a two-resonator entangled coherent state. Numerical simulation shows that the high-fidelity implementation of the phase gate and creation of the entangled coherent state are feasible with current circuit QED technology.
pacs:03.67.Lx, 42.50.Pq, 85.25.Cp, 42.50.Dv
Circuit quantum electrodynamics (QED), consisting of microwave resonators and superconducting (SC) qubits, is an analogue of cavity QED and a well-established platform for the investigation of light-matter interaction at the quantum level 04Blais (); s2 (); s3 (); s4 (). SC qubits are promising candidates to achieve scalable quantum computing and quantum information processing (QIP), especially with continuing improvements to coherence times s6 (); s7 (); s8 (); s9 (); s10 (); s11 (); s12 (); Pop (); s13 (). Furthermore, improving the quality factor of SC resonators is a key development for QIP. For example, a SC coplanar waveguide resonators with a loaded quality factor s40 (); s41 () or with internal quality factors above one million () have been previously reported s42 (). Recently, a SC microwave resonator with a loaded quality factor has been demonstrated in experiments s43 (). Combined with the large electric dipole moment of SC qubit, the strong-coupling or ultrastrong-coupling regime of a SC qubit with a resonator has been reported in experiments 04Wallraff (); 04Chiorescu (); 10Forniaz (); 10Niemczyk ().
The microwave resonators have been considered as good memory elements in QIP s43 (); M. Mariantoni (). Based on circuit QED, there is much interest in large-scale QIP which usually involves two or more microwave resonators/cavities. Recently, a number of proposals have been presented for synthesizing entangled states (e.g., Bell states, GHZ states, NOON states, and entangled coherent states) of two or more than two resonators 08Mariantoni (); 10Strauch (); njpMerkel (); 11Hu (); 12Strauch (); 12yang (); 13yang (); 14Su (); 15Xiong (); 16Sharma (); 16Zhao (); Z. Li (), and realizing two-qubit or multiqubit quantum gates with microwave photons distributed in two resonators 11Strauch (); M. Hua (). In addition, a great deal of quantum effects and operations involved multiple microwave resonators have been experimentally demonstrated in circuit QED. For instance, Ref. M. Mariantoni () experimentally implemented the quantum von Neumann architecture with SC circuits, Ref. H. Wang () generated photon NOON states of two SC resonators, Ref. L. Steffen () realized full deterministic quantum teleportation with feed-forward, and Ref. C. Wang () created a two-mode cat state of microwave fields in two SC cavities, respectively. Those progresses in circuit QED provide a promising perspective of microwave photons as resource for quantum communication and computation.
Cross-Kerr nonlinearity, known as cross phase modulation, is one of the most promising tool for quantum computation and QIP. In quantum optics, when two photons are simultaneously input into a nonlinear medium, the output photons undergo nonlinear optical effects named the cross-Kerr nonlinearity effect. During the last several decades, many QIP tasks have been proposed in optical systems based on cross-Kerr nonlinearity, including construction of quantum phase gates Q. A. Turchette (); K. Nemoto (), generation of macroscopic quantum superposition states C. C. Gerry (); H. Jeong (), completion of quantum teleportation D. Vitali (), realization of the quantum-nondemolition (QND) measurements N. Imoto (); P. Grangier (), and implementation of entanglement purification Y. B. Sheng ().
On the other hand, many theories and experiments involved cross-Kerr effect have been discussed in circuit QED S. Rebi (); O. Suchoi (); S. Kumar (); F. R. Ong (); G. Kirchmair (); B. Fan (); I. Hoi (); Y. Hu (); E. T. Holland (); H. Zhang (). Experimentally, based on the cross-Kerr nonlinearity, the observation of quantum state collapse and revival G. Kirchmair (), the investigation of the feasibility of microwave photon counting B. Fan (), and the realization of the giant cross-Kerr effect for propagating microwaves I. Hoi () have been reported. In recent years, the cross-Kerr nonlinearity between two microwave resonators has been extensively researched in circuit QED Y. Hu (); H. Zhang (); E. T. Holland (). For example, Refs. Y. Hu (); H. Zhang () proposed a scheme for implementing cross-Kerr nonlinearity between two SC resonators via an -type SC artificial atomic systems, and Ref. E. T. Holland () experimentally demonstrated a state dependent shift between two microwave cavities via a cross-Kerr effect. The -type four-level nature atoms was studied in Refs. H. Schmidt (); H. Kang (); Y. F. Chen (), and the Refs. Y. Hu (); H. Zhang () extended it to the SC circuits which consists of two SC transmon qubits coupled by a SQUID (quantum interference device). The Hamiltonian for a cross-Kerr interaction between two resonators is given by (in units of )
where is the coupling strength and () is the photon number operator of resonator ().
In this paper, we propose a method to realize a cross-Kerr nonlinearity interaction between two microwave resonators by coupling a three-level SC artificial atom [Fig. 1(a)]. This proposal has the following features and advantages: (i) Different from the previous works Y. Hu (); H. Zhang (), in our scheme only one operational step is needed, only one single three-level qutrit is used, and no need to use classical pulses. Thus the operation and experimental setup are greatly simplified. (ii) Because resonator photons are virtually excited and the qutrit is unexcited during the entire process, the effect of resonator decay, the unwanted inter-resonator crosstalk, and the qutrit decoherence are greatly minimized. (iii) This proposal can be applied to accomplish the same task with various SC qutrits (e.g., SC charge qutrits, transmon qutrits, Xmon qutrits, phase qutrits) coupled to two 1D resonators or two 3D cavities. (iv) Numerical simulation shows that our cross-Kerr interaction Hamiltonian can be used to high-fidelity realize a two-resonator qubits controlled-phase gate and generate a two-resonator entangled coherent state.
This paper is organized as follows. In Sec. II, we show how to realize a cross-Kerr interaction effect between two SC resonators in circuit QED. In Sec. III, we show how to use our effective Hamiltonian to construct a controlled-phase gate on two resonators, and then discuss how to create a macroscopic entangled coherent state of two resonators. In Sec. IV, we discuss the possible experimental implementation of our proposal and numerically calculate the operational fidelity for realizing a controlled-phase gate and generating a entangled coherent state. A concluding summary is given in Sec. V.
Ii Cross-Kerr nonlinearity effect in circuit QED
Consider a system consists of two SC coplanar waveguide resonators connected by a ladder-type SC flux qutrit (coupler) [Fig. 1(a)]. As shown in Fig. 1 (a,b), resonator () is off-resonantly coupled to the () transition of qutrit with a coupling constant (). In the interaction picture, the Hamiltonian of the whole system can be written as (in units of )
where and , () is a negative detuning but ( is a positive detuning. Here, () is the () transition frequency of qutrit and () is the frequency of resonator ().
Consider the large-detuning conditions and , the Hamiltonian (2) becomes s35 ()
where , , and . Under the large-detuning conditions and , the effective Hamiltonian changes to
where () is the cross-Kerr interaction coefficient. When the levels and are not occupied, the effective Hamiltonian (4) reduces to
where and are the photon number operators for resonators and . When the qutrit is in the state , the Hamiltonian describes the photon-number-dependent shift of the resonator , the Hamiltonian describes the cross-Kerr photon-photon interaction between the resonators and .
From Eq. (6), one can see that , thus, we obtain the effective cross-Kerr interaction Hamiltonian
It should be noted that the Hamiltonian (7) is different from the well-known cross-Kerr Hamiltonian describing the cross-Kerr interaction of two resonators with coefficient . It is because that the Hamiltonian (7) contains a coupler operator which is not involved in . In the next section, we first show how to use the effective Hamiltonian (7) to construct a controlled-phase gate on two resonators, and then discuss how to use the effective Hamiltonian (7) to create a macroscopic entangled coherent state of two resonators.
Iii Controlled-phase gate implementation and entangled-state preparation
Assume that the resonator () is in an arbitrary pure state () and the coupler is in the state . In this section, we first show how to use the Hamiltonian (5) or (7) to construct a two-qubit controlled-phase gate of two resonators and . We then discuss how to generate a macroscopic entangled coherent state for two resonators with Hamiltonian (5) or (7).
iii.1 Controlled-phase gate on two resonators
Suppose that the resonators and are in an arbitrary superposition state, respectively. Assume that and . Here, , , and are the normalized complex numbers; or ( or ) represents the vacuum state or the single photon state of resonator (). The time-evolution operator for the Hamiltonian (5) is defined as . Therefore, under the Hamiltonian (5), the resonator-system state evolves into
where . If we choose the interaction time , one can obtain a two-resonator qubits controlled-phase gate
where we have set ( is a positive integer). For and , one has the following relationship between the parameters
Notice that the effective coupling strength and the coupling strength can be adjusted by varying the detuning or . The level spacings of the SC “artificial atom” can be rapidly adjusted by varying external control parameters (e.g., magnetic flux applied to the superconducting loop of a superconducting phase, transmon, Xmon or flux qubit; see, e.g., s8 (); 08M. Neeley (); 09P. J. Leek (); 13J. D. Strand ()).
The controlled-phase gate of Eq. (9) is one of the paradigmatic gates for quantum information and quantum computation. By using it with single-qubit gates, a set of universal gates can be constructed. Hitherto, a great deal of theoretical proposals have been presented for realizing two-qubit controlled-phase gate in many physical systems. In circuit QED, a two SC qubits controlled-phase gate has already been experimentally demonstrated 12Fedorov (); s45 (); s46 (); s47 (). In addition, the controlled-phase gate with two microwave-photon-resonator qubits has been previously proposed in Refs. 11Strauch (); M. Hua (). The proposals 11Strauch (); M. Hua () require several operational steps and the application of classical pulses. Compared with Refs. 11Strauch (); M. Hua (), our proposal is much improved because our phase-gate can be achieved only using a single-step operation and no pulses are needed.
iii.2 Creation of a two-resonator macroscopic entangled coherent state
Assume that the resonators and are in coherent states and , respectively. Under the Hamiltonian (7), the joint state of the resonators evolves into
When the evolution time is equal to , one has . We divide the sum of Eq. (11) into a part with even and another with odd. It is apparent that an even/odd coherent state of can be expressed as
On substituting Eqs. (12) and (13) into Eq. (11), one obtains a macroscopic entangled coherent state
After returning to the original interaction picture by performing a unitary transformation , the entangled coherent state (14) becomes
where . Note that the entangled coherent states have many applications in the field of quantum information. For instance, they can be used to construct quantum gates s48 () (using coherent states as the logical qubits s49 ()), implement quantum key distribution s50 (), build quantum repeaters s51 (), test violation of Bell inequalities s52 (); s53 (), and applicant in quantum metrology s54 ().
Iv Possible experimental implementation
When the inter-resonator crosstalk is taken into account, the Hamiltonian (2) becomes , where describes the unwanted inter-resonator crosstalk, given by , with the inter-resonator crosstalk coupling strength and the two-resonator frequency detuning .
Including the dissipation and the dephasing, the dynamics of the lossy system is determined by the following master equation
where is given above, , , and with Here, () is the photon decay rate of resonator (). In addition, is the energy relaxation rate of the level of qutrit, () is the energy relaxation rate of the level of qutrit for the decay path (), and is the dephasing rate of the level of qutrit ().
The fidelity of the operation is given by
where is the output state of an ideal system (i.e., without dissipation, dephasing, and crosstalk); while is the final density operator of the system when the operation is performed in a realistic situation.
iv.1 Fidelity for the two-resonator qubits controlled-phase gate
As an example, we will consider the case of . In this case, we have the qutrit-resonator-system initially state and the output state . Here, is given by , which is obtained based on Eq. (8) for .
By solving the master equation (16), the fidelity of the gate operation can be calculated based on Eq (17). We now numerically calculate the fidelity for the operation. The frequency of SC resonators typically is 1 to 10 GHz. Thus, we choose GHz and GHz such that GHz. We then choose GHz and MHz in our scheme. In the following, we set the inter-resonator crosstalk strength , which can be readily met in experiments 12yang (). Furthermore, we set =1, , , , and .
Fig. 2 shows the fidelity versus . Green squares are plotted by choosing and , which correspond to the case that the systematic dissipation and dephasing are taken into account. Here we consider a rather conservative case for the decoherence times of flux qutrit Pop (); s13 (). It is noted that by designing the flux qutrit, the dipole matrix element can be made much smaller than that of the and transitions. Thus, . In addition, the leakage errors of a SC qutrit have been efficiently reduced in experiments Z. J. Chen ().
From Fig. 2, one can see that for GHz, a high fidelity 99.4% is achievable for a two-resonator qubits controlled-phase gate. For GHz, one has 342 MHz. The values of and are readily available in experiments 10Niemczyk (). In addition, one can obtain GHz and GHz. The transition frequency between two neighbor levels of a SC flux qutrit is typically in the range of 1-30 GHz. With the above given parameters, we obtain the large cross-Kerr interaction coefficient MHz. Our numerical simulation indicates that the high-fidelity implementation of a controlled-phase gate on two resonators is feasible with current circuit QED technology. As in Fig. 2, the effect of the qutrit decoherence and resonator decay on the fidelity is very small with the current parameter values. To illustrate the effect of the dissipation and dephasing of the system, we employ shorter qutrit decoherence and resonator-decay times in Fig. 3.
Fig. 3 displays the fidelity versus and , which is plotted by choosing GHz. From Fig. 4, one can obtain : (i) ; (ii) ; (iii) ; (iv) ; and (v) . Fig. 3 shows that for and , the fidelity can be greater than . This is because the qutrit is unexcited and resonator photons are virtually excited during the entire process, qutrit decoherence and resonator decay can be efficiently suppressed.
For the resonator frequencies and the resonator-decay times used in the Fig. 2, the required quality factors of the two resonators are and , which are attainable in experiments because a quality factor for SC coplanar waveguide resonators have been experimentally demonstrated s40 (); s41 (). For Fig. 3, the required quality factors for the resonators are and . Fig. 3 shows that the phase-gate operation also can be high-fidelity performed assisted by the low- resonators.
iv.2 Fidelity for generation of a two-resonator entangled coherent state
The fidelity for the operation is calculated based on Eq (17), where the ideal output state is and is obtained by numerically solving the master equation (16) for an initial input state . Here, state is given by Eq. (15).
We choose GHz, GHz, and GHz. We set MHz and MHz because the coupling strengths of values and are readily achievable in experiments 10Niemczyk (). In addition, we set and . The qutrit decoherence and the resonator-decay times used here are referred to the Fig. 3.
We now numerically calculate the fidelity for creation of the two-resonator entangled coherent state. In our numerical calculation, we consider the first terms in the expansions of coherent states () and (). Figure 4 shows the fidelity versus the normalized detuning with and , which is plotted for . For , a high fidelity 96.75 %, 97.03 %, 97.08 %, 97.09 % can be obtained for 4, 5, 6, and 7, respectively. For , we have GHz and MHz.
For resonators and with frequencies and dissipation times used in the Fig. 4, the quality factors of the two resonators are and . The numerical simulation indicates that the high-fidelity generation of a two-resonator entangled coherent state is feasible with current circuit QED technology.
We have proposed a method for realizing the cross-Kerr nonlinear interaction between two microwave resonators induced by a superconducting flux qutrit. Our present proposal differs from the previous protocols Y. Hu (); H. Zhang (). First, in our proposal only one qutrit is needed, thus the circuit complexity is much reduced. Second, there is only need one operation step and unnecessary to employ classical pulse, so the operation procedure is greatly simplified. Finally, due to the resonator photons are virtually excited and the coupler is unexcited for the entire process, the effect of resonator decay, the unwanted inter-resonator crosstalk, and the coupler decoherence are greatly minimized.
Although we assume that the cross-Kerr nonlinearity effect is performed between two SC coplanar waveguide resonators, using the three-level flux qutrit, our proposal can in principle also be applied to other solid devices, for example, the schemes based on other kinds of SC qutrits (e.g., SC charge qutrits, transmon qutrits, Xmon qutrits, phase qutrits) coupled to two 1D resonators or two 3D SC cavities, or based on the two nitrogen-vacancy center ensembles (behaves as two bosonic modes) coupled to a flux qutrit.
Based on our cross-Kerr interaction Hamiltonian, we implement a two-resonator qubits controlled-phase gate and generate a two-resonator entangled coherent state. Numerical simulation shows that the high-fidelity implementation of the phase gate and creation of the entangled coherent state are feasible with state-of-the-art circuit QED technology. Our finding provides a new way for realizing the cross-Kerr nonlinearity interaction between two microwave resonators, and such cross-Kerr effect may find applications in quantum information processing.
This work was supported by the Ministry of Science and Technology of China under Grant No. 2016YFA0301802, the National Natural Science Foundation of China under Grants No. 11375036 and No. 11175033, the Xinghai Scholar Cultivation Plan, and the Fundamental Research Funds for the Central Universities under Grants No. DUT15LK35 and No. DUT15TD47.
- (1) A. Blais, R. S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation, Phys. Rev. A 69, 062320 (2004).
- (2) J. Clarke and F. K. Wilhelm, Superconducting quantum bits, Nature 453, 1031 (2008).
- (3) J. Q. You and F. Nori, Atomic physics and quantum optics using superconducting circuits, Nature 474, 589 (2011).
- (4) I. Buluta, S. Ashhab, and F. Nori, Natural and artificial atoms for quantum computation, Rep. Prog. Phys. 74, 104401 (2011).
- (5) J. M. Chow, J. M. Gambetta, A. D. Córcoles, S. T. Merkel, J. A. Smolin, C. Rigetti, S. Poletto, G. A. Keefe, M. B. Rothwell, J. R. Rozen, M. B. Ketchen, and M. Steffen, Universal quantum gate set approaching fault-tolerant thresholds with superconducting qubits, Phys. Rev. Lett. 109, 060501 (2012).
- (6) J. B. Chang, M. R. Vissers, A. D. Corcoles, M. Sandberg, J. Gao, D. W. Abraham, J. M. Chow, J. M. Gambetta, M. B. Rothwell, G. A. Keefe, M. Steffen, and D. P. Pappas, Improved superconducting qubit coherence using titanium nitride, Appl. Phys. Lett. 103, 012602 (2013).
- (7) R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Y. Mutus, C. Neill, P. J. J. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, Coherent Josephson qubit suitable for scalable quantum integrated circuits, Phys. Rev. Lett. 111, 080502 (2013).
- (8) J. M. Chow, J. M. Gambetta, E. Magesan, D. W. Abraham, A. W. Cross, B. R. Johnson, N. A. Masluk, C. A. Ryan, J. A. Smolin, S. J. Srinivasan, and M. Steffen, Implementing a strand of a scalable fault-tolerant quantum computing fabric, Nature Comm. 5, 4015 (2014).
- (9) Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, Qubit architecture with high coherence and fast tunable coupling, Phys. Rev. Lett. 113, 220502 (2014).
- (10) M. Stern, G. Catelani, Y. Kubo, C. Grezes, A. Bienfait, D. Vion, D. Esteve, and P. Bertet, Flux qubits with long coherence times for hybrid quantum circuits, Phys. Rev. Lett. 113, 123601 (2014).
- (11) M. J. Peterer, S. J. Bader, X. Jin, F. Yan, A. Kamal, T. J. Gudmundsen, P. J. Leek, T. P. Orlando, W. D. Oliver, and S. Gustavsson, Coherence and decay of higher energy levels of a superconducting transmon qubit, Phys. Rev. Lett. 114, 010501 (2015).
- (12) I. M. Pop, K. Geerlings, G. Catelani, R. J. Schoelkopf, L. I. Glazman, and M. H. Devore, Coherent suppression of electromagnetic dissipation due to superconducting quasiparticles, Nature 508, 369 (2014).
- (13) F. Yan, S. Gustavsson, A. Kamal, J. Birenbaum, A. P. Sears, D. Hover, T. J. Gudmundsen, D. Rosenberg, G. Samach, S. Weber, J. L. Yoder, T. P. Orlando, J. Clarke, A. J. Kerman, and W. D. Oliver, The flux qubit revisited to enhance coherence and reproducibility, Nature Comm. 7, 12964 (2016).
- (14) W. Chen, D. A. Bennett, V. Patel, and J. E. Lukens, Substrate and process dependent losses in superconducting thin film resonators, Supercond. Sci. Technol. 21, 075013 (2008).
- (15) P. J. Leek, M. Baur, J. M. Fink, R. Bianchetti, L. Steffen, S. Filipp, and A. Wallraff, Cavity quantum electrodynamics with separate photon storage and qubit readout modes, Phys. Rev. Lett. 104, 100504 (2010).
- (16) A. Megrant, C. Neill, R. Barends, B. Chiaro, Y. Chen, L. Feigl, J. Kelly, E. Lucero, M. Mariantoni, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Y. Yin, J. Zhao, C. J. Palmstrøm, J. M. Martinis, and A. N. Cleland, Planar superconducting resonators with internal quality factors above one million, Appl. Phys. Lett. 100, 113510 (2012).
- (17) M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, Quantum memory with millisecond coherence in circuit QED, Phys. Rev. B 94, 014506 (2016).
- (18) A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics, Nature 431, 162 (2004).
- (19) I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij, Coherent dynamics of a flux qubit coupled to a harmonic oscillator, Nature 431, 159 (2004).
- (20) P. Forn-Díaz, J. Lisenfeld, D. Marcos, J. J. García-Ripoll, E. Solano, C. J. P. M. Harmans, and J. E. Mooij, Observation of the Bloch-Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime, Phys. Rev. Lett. 105, 237001 (2010).
- (21) T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, Circuit quantum electrodynamics in the ultrastrong-coupling regime, Nat. Physics 6, 772 (2010).
- (22) M. Mariantoni, H. Wang, T. Yamamoto, M. Neeley, R. C. Bialczak, Y. Chen, M. Lenander, E. Lucero, A. D. O’connell, D. Sank, M. Weides, J. Wenner, Y. Yin, J. Zhao, A. N. Korotkov, A. N. Cleland, J. M. Martinis, Implementing the Quantum von Neumann architecture with superconducting circuits, Science 334, 61 (2011).
- (23) M. Mariantoni, F. Deppe, A. Marx, R. Gross, F. K. Wilhelm, and E. Solano, Two-resonator circuit quantum electrodynamics: A superconducting quantum switch, Phys. Rev. B 78, 104508 (2008).
- (24) F. W. Strauch, K. Jacobs, and R. W. Simmonds, Arbitrary control of entanglement between two superconducting resonators, Phys. Rev. Lett. 105, 050501 (2010).
- (25) S. T. Merkel and F. K. Wilhelm, Generation and detection of NOON states in superconducting circuits, New J. Phys. 12, 093036 (2010).
- (26) Y. Hu and L. Tian, Deterministic generation of entangled photons in superconducting resonator arrays, Phys. Rev. Lett. 106, 257002 (2011).
- (27) F. W. Strauch, D. Onyango, K. Jacobs, and R. W. Simmonds, Entangled-state synthesis for superconducting resonators, Phys. Rev. A 85, 022335 (2012).
- (28) C. P. Yang, Q. P. Su, and S. Han, Generation of Greenberger-Horne-Zeilinger entangled states of photons in multiple cavities via a superconducting qutrit or an atom through resonant interaction, Phys. Rev. A 86, 022329 (2012).
- (29) C. P. Yang, Q. P. Su, S. B. Zheng, and S. Han, Generating entanglement between microwave photons and qubits in multiple cavities coupled by a superconducting qutrit, Phys. Rev. A 87, 022320 (2013).
- (30) Q. P. Su, C. P. Yang, and S. B. Zheng, Fast and simple scheme for generating NOON states of photons in circuit QED, Sci. Rep. 4, 3898 (2014).
- (31) S. J. Xiong, Z. Sun, J. M. Liu, T. Liu, and C. P. Yang, Efficient scheme for generation of photonic NOON states in circuit QED, Opt. Lett. 40, 2221 (2015).
- (32) R. Sharma and F. W. Strauch, Quantum state synthesis of superconducting resonators, Phys. Rev. A 93, 012342 (2016).
- (33) Y. J. Zhao, C. Q. Wang, X. B. Zhu, and Y. X. Liu, Engineering entangled microwave photon states through multiphoton interactions between two cavity fields and a superconducting qubit, Sci. Rep. 6, 23646 (2016).
- (34) Z. Li, S. L. Ma, Z. P. Yang, A. P. Fang, P. B. Li, S. Y. Gao, and F. L. Li, Generation and replication of continuous-variable quadripartite cluster and Greenberger-Horne-Zeilinger states in four chains of superconducting transmission line resonators, Phys. Rev. A 93, 042305 (2016).
- (35) F. W. Strauch, Quantum logic gates for superconducting resonator qudits, Phys. Rev. A 84, 052313 (2011).
- (36) M. Hua, M. J. Tao, and F. G. Deng, Universal quantum gates on microwave photons assisted by circuit quantum electrodynamics, Phys. Rev. A 90, 012328 (2014).
- (37) H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, Deterministic entanglement of photons in two superconducting microwave resonators, Phys. Rev. Lett. 106, 060401 (2011).
- (38) L. Steffen, Y. Salathe, M. Oppliger, P. Kurpiers, M. Baur, C. Lang, C. Eichler, G. Puebla-Hellmann, A. Fedorov, and A. Wallraff, Deterministic quantum teleportation with feed-forward in a solid state system, Nature 500, 319 (2013).
- (39) C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, R. J. Schoelkopf, A Schröinger cat living in two boxes, Science 352, 1087 (2016).
- (40) Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, Measurement of conditional phase shifts for quantum logic, Phys. Rev. Lett. 75, 4710 (1995).
- (41) K. Nemoto and W. J. Munro, Nearly deterministic linear optical controlled-NOT gate, Phys. Rev. Lett. 93, 250502 (2004).
- (42) C. C. Gerry, Generation of optical macroscopic quantum superposition states via state reduction with a Mach-Zehnder interferometer containing a Kerr medium, Phys. Rev. A 59, 4095 (2000).
- (43) H. Jeong, Using weak nonlinearity under decoherence for macroscopic entanglement generation and quantum computation, Phys. Rev. A 72, 034305 (2005).
- (44) D. Vitali, M. Fortunato, and P. Tombesi, Complete quantum teleportation with a Kerr nonlinearity, Phys. Rev. Lett. 85, 445 (2000).
- (45) N. Imoto, H. A. Haus, and Y. Yamamoto, Quantum nondemolition measurement of the photon number via the optical Kerr effect, Phys. Rev. A 32, 2287 (1985).
- (46) P. Grangier, J. A. Levenson, and J. P. Poizat, Quantum non-demolition measurements in optics, Nature 396, 537 (1998).
- (47) Y. B. Sheng, F. G. Deng, and H. Y. Zhou, Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-Kerr nonlinearity, Phys. Rev. A 77, 042308 (2008).
- (48) S. Rebić, J. Twamley, and G. J. Milburn, Giant Kerr nonlinearities in circuit quantum electrodynamics, Phys. Rev. Lett. 103, 150503 (2009).
- (49) O. Suchoi, B. Abdo, E. Segev, O. Shtempluck, M. P. Blencowe, and Eyal Buks, Intermode dephasing in a superconducting stripline resonator, Phys. Rev. B 81, 174525 (2010).
- (50) S. Kumar and D. P. DiVincenzo, Exploiting Kerr cross nonlinearity in circuit quantum electrodynamics for nondemolition measurements, Phys. Rev. B 82, 014512 (2010).
- (51) F. R. Ong, M. Boissonneault, F. Mallet, A. Palacios-Laloy, A. Dewes, A. C. Doherty, A. Blais, P. Bertet, D. Vion, and D. Esteve, Circuit QED with a nonlinear resonator: ac-Stark shift and dephasing, Phys. Rev. Lett. 106, 167002 (2011).
- (52) G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Observation of quantum state collapse and revival due to the single-photon Kerr effect, Nature 495, 205 (2013).
- (53) B. Fan, A. F. Kockum, J. Combes, G. Johansson, I. Hoi, C. M. Wilson, P. Delsing, G. J. Milburn, and T. M. Stace, Breakdown of the cross-Kerr scheme for photon counting, Phys. Rev. Lett. 110, 053601 (2013).
- (54) I. Hoi, A. F. Kockum, T. Palomaki, T. M. Stace, B. Fan, L. Tornberg, S. R. Sathyamoorthy, G. Johansson, P. Delsing, and C. M. Wilson, Giant cross-Kerr effect for propagating microwaves induced by an artificial atom, Phys. Rev. Lett. 111, 053601 (2013).
- (55) Y. Hu, G. Q. Ge, S. Chen, X. F. Yang, and Y. L. Chen, Cross-Kerr-effect induced by coupled Josephson qubits in circuit quantum electrodynamics, Phys. Rev. A 84, 012329 (2011).
- (56) H. Zhang and H. Wang, Entanglement concentration of microwave photons based on Kerr effect in circuit QED, arXiv:1602.08919 (2016).
- (57) E. T. Holland, B. Vlastakis, R. W. Heeres, M. J. Reagor, U. Vool, Z. Leghtas, L. Frunzio, G. Kirchmair, M. H. Devoret, M. Mirrahimi, and R. J. Schoelkopf, Single-photon-resolved Cross-Kerr interaction for autonomous stabilization of photon-number states, Phys. Rev. Lett. 115, 180501 (2015).
- (58) H. Schmidt and A. Imamoglu, Giant Kerr nonlinearities obtained by electromagnetically induced transparency, Opt. Lett. 21, 1936 (1996).
- (59) H. Kang and Y. Zhu, Observation of large Kerr nonlinearity at low light intensities, Phys. Rev. Lett. 91, 093601 (2003).
- (60) Y. F. Chen, C. Y. Wang, S. H. Wang, and I. Yu, Low-light-level cross-phase-modulation based on stored light pulses, Phys. Rev. Lett. 96, 043603 (2006).
- (61) D. F. James and J. Jerke, Effective Hamiltonian theory and its applications in quantum information, Can. J. Phys. 85, 625 (2007).
- (62) M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, A. O’Connell, H. Wang, A. N. Cleland, and J. M. Martinis, Process tomography of quantum memory in a Josephson-phase qubit coupled to a two-level state, Nat. Physics 4, 523 (2008).
- (63) P. J. Leek, S. Filipp, P. Maurer, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, and A. Wallraff, Using sideband transitions for two-qubit operations in superconducting circuits, Phys. Rev. B 79, 180511 (2009).
- (64) J. D. Strand, M. Ware, F. Beaudoin, T. A. Ohki, B. R. Johnson, A. Blais, and B. L. T. Plourde, First-order sideband transitions with flux-driven asymmetric transmon qubits, Phys. Rev. B 87, 220505 (2013).
- (65) A. Fedorov, L. Steffen, M. Baur, M. P. da Silva£¬and A. Wallraff, Implementation of a Toffoli gate with superconducting circuits, Nature 481, 170 (2012).
- (66) J. H. Plantenberg, P. C. de Groot, C. J. P. M. Harmans, and J. E. Mooij, Demonstration of controlled-NOT quantum gates on a pair of superconducting quantum bits, Nature 447, 836 (2007).
- (67) L. DiCarlo, J. M. Chow, J. M. Gambetta, Lev S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Demonstration of two-qubit algorithms with a superconducting quantum processor, Nature 460, 240 (2009).
- (68) R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, Superconducting quantum circuits at the surface code threshold for fault tolerance, Nature 508, 500 (2014).
- (69) H. Jeong and M. S. Kim, Efficient quantum computation using coherent states, Phys Rev. A 65, 042305 (2002).
- (70) T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, Quantum computation with optical coherent states, Phys. Rev. A 68, 042319 (2003).
- (71) F. Grosshans and P. Grangier, Continuous variable quantum cryptography using coherent states, Phys. Rev. Lett. 88, 057902 (2002).
- (72) P. V. Loock, N. Lütkenhaus, W. J. Munro, and K. Nemoto, Quantum repeaters using coherent-state communication, Phys. Rev. A 78, 062319 (2008).
- (73) A. Gilchrist, P. Deuar, and M. D. Reid, Contradiction of quantum mechanics with local hidden variables for quadrature phase amplitude measurements, Phys. Rev. Lett. 80, 3169 (1998).
- (74) A. Gilchrist, P. Deuar, and M. D. Reid, Contradiction of quantum mechanics with local hidden variables for quadrature phase measurements on pair-coherent states and squeezed macroscopic superpositions of coherent states, Phys. Rev. A 60, 4259 (1999).
- (75) J. Joo, W. J. Munro, and T. P. Spiller, Quantum metrology with entangled coherent states, Phys. Rev. Lett. 107, 083601 (2011).
- (76) Z. J. Chen, J. Kelly, C. Quintana, R. Barends, B. Campbell, Y. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Lucero, E. Jeffrey, A. Megrant, J. Mutus, M. Neeley, C. Neill, P. J. J. O’Malley, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, T. C. White, A. N. Korotkov, and J. M. Martinis, Measuring and suppressing quantum state leakage in a superconducting qubit, Phys. Rev. Lett. 116, 020501 (2016).