Abstract

Large optical nonlinearities can create fancy physics, such as big Schrödinger-cat states and quadrature squeezing. We present the possibility to practically generate macroscopic Schrödinger-cat states, based on a giant Kerr nonlinearity, in a diamond nitrogen-vacancy ensemble interacting with two coupled flux-qubits. The nonlinearity comes from a four-level N-type configuration formed by two coupled flux-qubits under the appropriately driving fields. We discuss the experimental feasibility in the presence of system dissipations using current laboratory technology and our scheme can be easily extended to other ensemble systems.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (3)

Y. Maleki and A. M. Zheltikov, “Generating maximally-path-entangled number states in two spin ensembles coupled to a superconducting flux qubit,” Phys. Rev. A 97, 012312 (2018).
[Crossref]

Y. Maleki and A. M. Zheltikov, “Witnessing quantum entanglement in ensembles of nitrogen-vacancy centers coupled to a superconducting resonator,” Opt. Express 14, 17849–17858 (2018).
[Crossref]

K. Cai, Z.-W. Pan, R.-X. Wang, D. Ruan, Z.-Q. Yin, and G.-L. Long, “Single phonon source based on a giant polariton nonlinear effect,” Opt. Lett. 43, 1163–1166 (2018).
[Crossref] [PubMed]

2017 (3)

H. Liu, M. B. Plenio, and J. Cai, “Scheme for detection of single-molecule radical pair reaction using spin in diamond,” Phys. Rev. Lett. 118, 200402 (2017).
[Crossref] [PubMed]

K. G. Johnson, J. D. Wong-Campos, B. Neyenhuis, J. Mizrahi, and C. Monroe, “Ültrafast creation of large Schrödinger cat states of an atom,” Nat. Commun. 8, 697 (2017).
[Crossref]

T. Astner, S. Nevlacsil, N. Peterschofsky, A. Angerer, S. Rotter, S. Putz, J. Schmiedmayer, and J. Majer, “Coherent coupling of remote spin ensembles via a cavity bus,” Phys. Rev. Lett. 118, 140502 (2017).
[Crossref] [PubMed]

2016 (6)

J.-Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116, 163602 (2016).
[Crossref] [PubMed]

W. L. Song, W. L. Yang, Z. Q. Yin, C. Y. Chen, and M. Feng, “Controllable quantum dynamics of inhomogeneous nitrogen-vacancy center ensembles coupled to superconducting resonators,” Sci. Rep. 6, 33271 (2016).
[Crossref] [PubMed]

M. Bruderer, P. F. Acebal, R. Aurich, and M. B. Plenio, “Sensing of single nuclear spins in random thermal motion with proximate nitrogen-vacancy centers,” Phys. Rev. B 93, 115116 (2016).
[Crossref]

W. L. Ma, S. S. Li, G. Y. Cao, and R. B. Liu, “Atomic-scale positioning of single spins via multiple nitrogen-vacancy centers,” Phys. Rev. Applied 5, 044016 (2016).
[Crossref]

A. Facon, E.-K. Dietsche, D. Grosso, S. Haroche, J.-M. Raimond, M. Brune, and S. Gleyzes, “A sensitive electrometer based on a Rydberg atom in a Schrödinger-cat state,” Nature 535, 262–265 (2016).
[Crossref] [PubMed]

T. Unden, P. Balasubramanian, D. Louzon, Y. Vinkler, M. B. Plenio, M. Markham, D. Twitchen, A. Stacey, I. Lovchinsky, A. O. Sushkov, M. D. Lukin, A. Retzker, B. Naydenov, L. P. McGuinness, and F. Jelezko, “Quantum metrology enhanced by repetitive quantum error correction,” Phys. Rev. Lett. 116, 230502 (2016).
[Crossref] [PubMed]

2015 (3)

J. Huang, X. Qin, H. Zhong, Y. Ke, and C. Lee, “Quantum metrology with spin cat states under dissipation,” Sci. Rep. 5, 17894 (2015).
[Crossref] [PubMed]

T. Wolf, P. Neumann, K. Nakamura, H. Sumiya, T. Ohshima, J. Isoya, and J. Wrachtrup, “Subpicotesla diamond magnetometry,” Phys. Rev. X 5, 041001 (2015).

L. P. Neukirch, E. von Haartman, J. M. Rosenholm, and A. N. Vamivakas, “Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond,” Nat. Photo. 9, 653–657 (2015).
[Crossref]

2014 (3)

A. Z. Chaudhry, “Ütilizing nitrogen-vacancy centers to measure oscillating magnetic fields,” Phys. Rev. A 90, 042104 (2014).
[Crossref]

X. Zhu, Y. Matsuzaki, R. Amsüss, K. Kakuyanagi, T. S. Oka, N. Mizuochi, K. Nemoto, K. Semba, W. J. Munro, and S. Saito, “Observation of dark states in a superconductor diamond quantum hybrid system,” Nat. Commun. 5, 3424 (2014).
[Crossref] [PubMed]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

2013 (5)

S. Saito, X. Zhu, R. Amsüss, Y. Matsuzaki, K. Kakuyanagi, T. S. Oka, N. Mizuochi, K. Nemoto, W. J. Munro, and K. Semba, “Towards realizing a quantum memory for a superconducting qubit: storage and retrieval of quantum states,” Phys. Rev. Lett. 111, 107008 (2013).
[Crossref]

Z.-L. Xiang, X.-Y. Lü, T.-F. Li, J. Q. You, and F. Nori, “Hybrid quantum circuit consisting of a superconducting flux qubit coupled to a spin ensemble and a transmission-line resonator,” Phys. Rev. B 87, 144516 (2013).
[Crossref]

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, 205–209 (2013).
[Crossref] [PubMed]

J. S. N. Nielsen, Y. Eto, C. W. Lee, H. Jeong, and M. Sasaki, “Quantum tele-amplification with a continuous-variable superposition state,” Nat. Photon. 7, 439–443 (2013).
[Crossref]

M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, and S. Bos, “Matter-wave interferometry of a levitated thermal nano-oscillator induced and probed by a spin,” Phys. Rev. Lett. 111, 180403 (2013).
[Crossref] [PubMed]

2012 (1)

W. L. Yang, Z. Yin, Z. X. Chen, S. Kou, M. Feng, and C. H. Oh, “Quantum simulation of an artificial Abelian gauge field using nitrogen-vacancy-center ensembles coupled to superconducting resonators,” Phys. Rev. A 86, 012307 (2012).
[Crossref]

2011 (5)

Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Drau, J.-F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Hybrid quantum circuit with a superconducting qubit coupled to a spin ensemble,” Phys. Rev. Lett. 107, 220501 (2011).
[Crossref] [PubMed]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107, 020405 (2011).
[Crossref] [PubMed]

M. Kira, S. W. Koch, R. P. Smith, A. E. Hunter, and S. T. Cundiff, “Quantum spectroscopy with Schrödinger-cat states,” Nat. Phys. 7, 799–804 (2011).
[Crossref]

X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nature 478, 221–224 (2011).
[Crossref] [PubMed]

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107, 083601 (2011).
[Crossref] [PubMed]

2010 (4)

B. R. Johnson, M. D. Reed, A. A. Houck, D. I. Schuster, L. S. Bishop, E. Ginossar, J. M. Gambetta, L. DiCarlo, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Quantum non-demolition detection of single microwave photons in a circuit,” Nat. Phys. 6, 663 (2010).
[Crossref]

G. Chen, H. Zhang, Y. Yang, R. Wang, L. Xiao, and S. Jia, “Qubit-induced high-order nonlinear interaction of the polar molecules in a stripline cavity,” Phys. Rev. A 82, 013601 (2010).
[Crossref]

D. Marcos, M. Wubs, J. M. Taylor, R. Aguado, M. D. Lukin, and A. S. Sørensen, “Coupling nitrogen-vacancy centers in diamond to superconducting flux qubits,” Phys. Rev. Lett. 105, 210501 (2010).
[Crossref]

W.-B. Gao, C.-Y. Lu, X.-C. Yao, P. Xu, O. Gühne, A. Goebel, Y.-A. Chen, C.-Z. Peng, Z.-B. Chen, and J.-W. Pan, “Experimental demonstration of a hyper-entangled ten-qubit Schrödinger cat state,” Nat. Phys. 6, 331–335 (2010).
[Crossref]

2009 (4)

R. Harris, F. Brito, A. J. Berkley, J. Johansson, M. W. Johnson, T. Lanting, P. Bunyk, E. Ladizinsky, B. Bumble, A. Fung, A. Kaul, A. Kleinsasser, and S. Han, “Synchronization of multiple coupled rf-SQUID flux qubits,” New J. Phys. 11, 123022 (2009).
[Crossref]

R. Harris, T. Lanting, A. J. Berkley, J. Johansson, M. W. Johnson, P. Bunyk, E. Ladizinsky, N. Ladizinsky, T. Oh, and S. Han, “Compound Josephson-junction coupler for flux qubits with minimal crosstalk,” Phys. Rev. B 80, 052506 (2009).
[Crossref]

S. Rebic, J. Twamley, and G. J. Milburn, “Giant Kerr nonlinearities in circuit quantum electrodynamics,” Phys. Rev. Lett. 103, 150503 (2009).
[Crossref] [PubMed]

M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, J. M. Martinis, and A. N. Cleland, “Synthesizing arbitrary quantum states in a superconducting resonator,” Nature 459, 546–549 (2009).
[Crossref] [PubMed]

2008 (1)

R. Harris, M. W. Johnson, S. Han, A. J. Berkley, J. Johansson, P. Bunyk, E. Ladizinsky, S. Govorkov, M. C. Thom, S. Uchaikin, B. Bumble, A. Fung, A. Kaul, A. Kleinsasser, M. H. S. Amin, and D. V. Averin, “Probing noise in flux qubits via macroscopic resonant tunneling,” Phys. Rev. Lett. 101, 117003 (2008).
[Crossref] [PubMed]

2007 (1)

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical Schrödinger cat from photon number states,” Nature 448, 784–786 (2007).
[Crossref] [PubMed]

2006 (1)

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2005 (3)

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M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
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2004 (2)

F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillations in a single electron spin,” Phys. Rev. Lett. 92, 076401 (2004).
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1999 (4)

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1985 (2)

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1980 (1)

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D. Marcos, M. Wubs, J. M. Taylor, R. Aguado, M. D. Lukin, and A. S. Sørensen, “Coupling nitrogen-vacancy centers in diamond to superconducting flux qubits,” Phys. Rev. Lett. 105, 210501 (2010).
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M. Bruderer, P. F. Acebal, R. Aurich, and M. B. Plenio, “Sensing of single nuclear spins in random thermal motion with proximate nitrogen-vacancy centers,” Phys. Rev. B 93, 115116 (2016).
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Figures (8)

Fig. 1
Fig. 1 (a) Schematic diagram of the hybrid quantum system composed of two coupled CJJ rf-SQUID qubits and an NVE. Inset: the level structure of an NV-center [30] under a weak classical field with an amplitude ξp coupling the states | ± to | 0 . (b) The N-type level structure for the coupled flux-qubits with ω = J + J 2 + Ω q 2 and ω ¯ = J 2 + Ω q 2 J . A control field with Rabi frequency Ω and frequency ω c = 2 J is in resonance with the states | 2 and | 3 . (c) Three-dimensional plot of log 10 [ g 2 ( 0 ) ] as functions of Ω and ϵp. The inset denotes the nonlinearity factor χ varying with the amplitude ϵp of the driving field along the gradient direction (red curve). Here the coupling strengths λ ¯ / κ = 400 and λ ˜ / κ = 200 . The decay paths are set as γ 41 = γ 32 = 0.1 γ and γ 31 = γ 21 = γ 42 = γ 43 = γ . Where, γ i j represent the decay rate from the states | i to | j . Other parameters are Δ = δ = κ and γ / κ = 0.4 .
Fig. 2
Fig. 2 (a) Time evolution of the fidelity F . (b) Wigner function W ( β ) at five different time points corresponding to t1, t2, t3, t4 and t5 in (a). (c) Probability distribution of the rotated quadrature operator PX at time points t3 (blue solid curve) and t4 (black dashed curve). Here we choose κ = 1 , γ = 0.4 , Δ = δ = 1 , α = 2 e i π / 4 , ϵ p = 2 , λ ˜ = 300 , Ω = 50 and decay paths are chosen same as in Fig. 1(c).
Fig. 3
Fig. 3 (a) Optimal fidelity of the cat state as a function of the cat’s size. Inset: the dots denote the oscillating period of the fidelity for | α | 2 5 and the line means a constant obtained by fitting these dots. (b) Optimal time c t τ op in variation with the cat’s size. Inset: the dots represents Δ τ op between the nearest-neighbor ladders and the curve denotes an exponential fitting. (c) Dynamical evolution of the fidelity for three sizes of the cat states around the inflection point | α | 2 = 5.05 , which is labeled as a red dot in (a). (d) Probability distribution of the rotated quadrature operator PX at time points r o t e c t κ τ op = 0.02 ( | α | 2 = 4.8 , 5 ) and 0.025 ( | α | 2 = 5.1 , 5.8 ). Here the parameters used are the same as in Fig. 2.
Fig. 4
Fig. 4 (a) and (b) Fidelities of the cat state prepared under different flux-qubit dissipation rates and NVE decay rates, where we set u 0 / 2 π = 0.1 MHz, and for convenience of presentation the parameters below are written in units of u0. We have κ = 1 in (a) and γ = 1 in (b). The solid, dashed and dotted curves in both (a) and (b) correspond to λ ˜ = 300 , 200 and 100 , respectively. Inset: time evolution of the fidelity with t λ ˜ = 300 . (c) Probability distribution of the rotated quadrature operator PX at the time point t4 of Fig. 2. The dotted, dashed and solid curves correspond to the excitation numbers n c = 1, 2 and 5. In (c), κ = 1 and γ = 0.4 . Inset: Wigner function W ( β ) with respect to n c = 1 and 5. Other parameters in (a-c) are p h a = 2 e i π / 4 , ϵ p = 2 , λ ¯ = 400 and Ω = 50 , respectively.
Fig. 5
Fig. 5 (a) Fidelities of the cat state prepared under different dephasing rates κd. (b) A detail dynamical evolution of the fidelities for three different dephasing rates at a coupling λ ˜ = 300 . (c) The Wigner function W ( β ) atthe optimal cat state position of in (b) for the case κ d = 0.5 . Here κ = 1 and γ = 0.4 and the other parameters are same as in Fig. 4.
Fig. 6
Fig. 6 Squeezing spectrum S ( ω ) under different conditions, where (a) the solid (dashed) curve denotes Ω = 30 (36) and (b) Ω = 50 (60) in unit of u0 with u 0 = 0.1 MHz. Here we choose Δ = δ = 5 and ϵ p = 2 . Other parameters are the same as in Fig. 1(c) of main text.
Fig. 7
Fig. 7 (a) Dynamical evolution of the fidelity for different sizes of the cat states. The curves from left to right correspond to the cat states with the size | α | 2 changing form 2 to 6.5. (b) and (c) are zooming-in plots of Figs. 3(a) and 3(b) of the main text, respectively.
Fig. 8
Fig. 8 (a) and (c) Optimal fidelity of the cat state as a function of the cat’s size. Inset: the dots denote the oscillating period of the fidelity for | α | 2 5 and the line means a linear fitting of these dots. (b) and (d) The time τ op for obtaining the optimal fidelity of the cat state. Inset: the dots represent the corresponding spacing Δ τ op between the nearest neighbor two ladders and the curve denotes an exponential fitting. The black, red and green data and curves in (a) and (b) correspond to γ = 0.2 , 0.4 and 1 with κ = 1 , and in (c) and (d) they are corresponding to κ = 0.5 , 1 and 2 with γ = 0.4 . The other parameters are same in Fig. 3 of main text.

Equations (11)

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H = Ω q 2 i = 1 2 σ x i + J σ z 1 σ z 2 + i = 1 M D S z , i 2 + i = 1 M S x , i ( g 1 σ z 1 + g 2 σ z 2 ) ,
H 1 = Δ | 4 4 | + δ | 3 3 | + i = 1 M [ | b i i 0 | ( g ¯ | 1 3 | + g ˜ | 2 4 | ) + h . c . ] ,
H 2 = H 0 + a ( λ ¯ | 3 1 | + λ ˜ | 4 2 | ) + h . c . ,
ρ ˙ = i [ H , ρ ] + κ 2 D [ a , ρ ] + j < k γ j , k 2 D [ σ j k , ρ ] ,
Δ | α | 2 C , Δ τ op = a 1 e a 2 n + a 3 ,
( | 4 | 3 | 2 | 1 ) = 1 2 ( cos  θ sin  θ sin  θ cos  θ 1 0 0 1 0 1 1 0 sin  θ cos  θ cos  θ sin  θ ) ( | e e | e g | g e | g g ) .
( | e e | e g | g e | g g ) = 1 2 ( cos  θ 1 0 sin  θ sin  θ 0 1 cos  θ sin  θ 0 1 cos  θ cos  θ 1 0 sin  θ ) ( | 4 | 3 | 2 | 1 ) .
σ z 1 = cos  θ | 4 3 | + sin  θ | 4 2 | + sin  θ | 3 1 | cos  θ | 2 1 | + h . c . , σ z 2 = cos  θ | 4 3 | sin  θ | 4 2 | + sin  θ | 3 1 | + cos  θ | 2 1 | + h . c
H 3 = Δ | 4 4 | + δ | 3 3 | + i = 1 M [ e i D t | b i i 0 | + h . c . ] [ G ¯ ( cos   θ e i ω ¯ t | 4 + sin   θ e i D t | | 1 ) 3 | + G ˜ ( sin θ e i D t | 4 cos θ e i ω ¯ t | 1 ) 2 | + h . c . ] ,
H 4 = Δ | 4 4 | + δ | 3 3 | + sin  θ i = 1 M [ | b i i 0 | ( G ¯ | 1 3 | + G ˜ | 2 4 | ) + h . c . ] ,
S ( ω ) = 2 ξ 2 ( 1 3 ξ 2 ) 2 + 16 ξ 2 ξ [ ( ω 2 / κ 2 + 1 3 ξ 2 ) ( 1 3 ξ 2 ) + 16 ξ 2 ] ( 1 3 ξ 2 ) 2 + 16 ξ 2 [ ( ω 2 / κ 2 + 1 3 ξ 2 ) 2 + 12 ξ 2 ]

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