Abstract

Recently, significant effort has been devoted to the study of atom–photon quantum interfaces using intracavity Rydberg-blocked atomic ensembles, which may serve as the platform for many essential quantum information processing tasks. In this paper, we use a theoretical analysis of this platform where the ground-Rydberg transition is realized by a two-photon transition, and we report our recent findings regarding the Jaynes–Cummings model on optical domain and robust atom–photon quantum gates. Our implementation with typical alkali atoms, such as Rb or Cs, requires an optical cavity of moderately high finesse and the condition that the cold atomic ensemble is well within the Rydberg-blockade radius. The analysis focuses on the atomic ensemble’s collective coupling to the quantized optical field in the cavity mode, and we demonstrate its capability to serve as a controlled-PHASE gate between photonic qubit and matter qubit, where the photonic qubit is endowed with a reasonably wide frequency tuning range. The detrimental effects associated with several major decoherence factors in this system are also considered in the analysis.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (3)

C. R. Murray, I. Mirgorodskiy, C. Tresp, C. Braun, A. Paris-Mandoki, A. V. Gorshkov, S. Hofferberth, and T. Pohl, “Photon subtraction by many-body decoherence,” Phys. Rev. Lett. 120, 113601 (2018).
[Crossref]

F. Motzoi and K. Mølmer, “Precise single-qubit control of the reflection phase of a photon mediated by a strongly-coupled ancilla-cavity system,” New J. Phys. 20, 053029 (2018).
[Crossref]

B.-Q. Ou, C. Liu, Y. Sun, and P.-X. Chen, “Deterministically swapping frequency-bin entanglement from photon-photon to atom-photon hybrid systems,” Phys. Rev. A 97, 023839 (2018).
[Crossref]

2017 (5)

D. Petrosyan, F. Motzoi, M. Saffman, and K. Mølmer, “High-fidelity Rydberg quantum gate via a two-atom dark state,” Phys. Rev. A 96, 042306 (2017).
[Crossref]

O. Lahad and O. Firstenberg, “Induced cavities for photonic quantum gates,” Phys. Rev. Lett. 119, 113601 (2017).
[Crossref]

C. R. Murray and T. Pohl, “Coherent photon manipulation in interacting atomic ensembles,” Phys. Rev. X 7, 031007 (2017).
[Crossref]

J. Lee, M. J. Martin, Y.-Y. Jau, T. Keating, I. H. Deutsch, and G. W. Biedermann, “Demonstration of the Jaynes-Cummings ladder with Rydberg-dressed atoms,” Phys. Rev. A 95, 041801 (2017).
[Crossref]

Y. Zeng, P. Xu, X. He, Y. Liu, M. Liu, J. Wang, D. J. Papoular, G. V. Shlyapnikov, and M. Zhan, “Entangling two individual atoms of different isotopes via Rydberg blockade,” Phys. Rev. Lett. 119, 160502 (2017).
[Crossref]

2016 (8)

M. Saffman, “Quantum computing with atomic qubits and Rydberg interactions: progress and challenges,” J. Phys. B 49, 202001 (2016).
[Crossref]

C. Tresp, C. Zimmer, I. Mirgorodskiy, H. Gorniaczyk, A. Paris-Mandoki, and S. Hofferberth, “Single-photon absorber based on strongly interacting Rydberg atoms,” Phys. Rev. Lett. 117, 223001 (2016).
[Crossref]

B. Hacker, S. Welte, G. Rempe, and S. Ritter, “A photon-photon quantum gate based on a single atom in an optical resonator,” Nature 536, 193–196 (2016).
[Crossref]

T. Keating, C. H. Baldwin, Y.-Y. Jau, J. Lee, G. W. Biedermann, and I. H. Deutsch, “Arbitrary Dicke-state control of symmetric Rydberg ensembles,” Phys. Rev. Lett. 117, 213601 (2016).
[Crossref]

J. Ningyuan, A. Georgakopoulos, A. Ryou, N. Schine, A. Sommer, and J. Simon, “Observation and characterization of cavity Rydberg polaritons,” Phys. Rev. A 93, 041802 (2016).
[Crossref]

S. Das, A. Grankin, I. Iakoupov, E. Brion, J. Borregaard, R. Boddeda, I. Usmani, A. Ourjoumtsev, P. Grangier, and A. S. Sørensen, “Photonic controlled-phase gates through Rydberg blockade in optical cavities,” Phys. Rev. A 93, 040303 (2016).
[Crossref]

A. C. J. Wade, M. Mattioli, and K. Mølmer, “Single-atom single-photon coupling facilitated by atomic-ensemble dark-state mechanisms,” Phys. Rev. A 94, 053830 (2016).
[Crossref]

Y. Wang, A. Kumar, T.-Y. Wu, and D. S. Weiss, “Single-qubit gates based on targeted phase shifts in a 3D neutral atom array,” Science 352, 1562–1565 (2016).
[Crossref]

2015 (7)

I. I. Beterov and M. Saffman, “Rydberg blockade, Förster resonances, and quantum state measurements with different atomic species,” Phys. Rev. A 92, 042710 (2015).
[Crossref]

M. Khazali, K. Heshami, and C. Simon, “Photon-photon gate via the interaction between two collective Rydberg excitations,” Phys. Rev. A 91, 030301 (2015).
[Crossref]

Y. M. Hao, G. W. Lin, K. Xia, X. M. Lin, Y. P. Niu, and S. Q. Gong, “Quantum controlled-phase-flip gate between a flying optical photon and a Rydberg atomic ensemble,” Sci. Rep. 5, 10005 (2015).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum-optical nonlinearities induced by Rydberg-Rydberg interactions: a perturbative approach,” Phys. Rev. A 92, 043841 (2015).
[Crossref]

A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Rev. Mod. Phys. 87, 1379–1418 (2015).
[Crossref]

M. Ebert, M. Kwon, T. G. Walker, and M. Saffman, “Coherence and Rydberg blockade of atomic ensemble qubits,” Phys. Rev. Lett. 115, 093601 (2015).
[Crossref]

K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr, L. Isenhower, and M. Saffman, “Rydberg-blockade controlled-not gate and entanglement in a two-dimensional array of neutral-atom qubits,” Phys. Rev. A 92, 022336 (2015).
[Crossref]

2014 (8)

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref]

I. I. Beterov, T. Andrijauskas, D. B. Tretyakov, V. M. Entin, E. A. Yakshina, I. I. Ryabtsev, and S. Bergamini, “Jaynes-Cummings dynamics in mesoscopic ensembles of Rydberg-blockaded atoms,” Phys. Rev. A 90, 043413 (2014).
[Crossref]

A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature 508, 237–240 (2014).
[Crossref]

C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, “Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects,” Phys. Rev. A 89, 022317 (2014).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

C. Liu, Y. Sun, L. Zhao, S. Zhang, M. M. T. Loy, and S. Du, “Efficiently loading a single photon into a single-sided Fabry–Perot cavity,” Phys. Rev. Lett. 113, 133601 (2014).
[Crossref]

D. Paredes-Barato and C. S. Adams, “All-optical quantum information processing using Rydberg gates,” Phys. Rev. Lett. 112, 040501 (2014).
[Crossref]

E. Bimbard, R. Boddeda, N. Vitrant, A. Grankin, V. Parigi, J. Stanojevic, A. Ourjoumtsev, and P. Grangier, “Homodyne tomography of a single photon retrieved on demand from a cavity-enhanced cold atom memory,” Phys. Rev. Lett. 112, 033601 (2014).
[Crossref]

2013 (2)

A. Reiserer, S. Ritter, and G. Rempe, “Nondestructive detection of an optical photon,” Science 342, 1349–1351 (2013).
[Crossref]

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

2012 (4)

S. Ritter, C. Nolleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mucke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012).
[Crossref]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Y. O. Dudin and A. Kuzmich, “Strongly interacting Rydberg excitations of a cold atomic gas,” Science 336, 887–889 (2012).
[Crossref]

T. C. V. Opatrný and K. Mølmer, “Spin squeezing and Schrödinger-cat-state generation in atomic samples with Rydberg blockade,” Phys. Rev. A 86, 023845 (2012).
[Crossref]

2011 (4)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. 83, 33–80 (2011).
[Crossref]

J. Honer, R. Löw, H. Weimer, T. Pfau, and H. P. Büchler, “Artificial atoms can do more than atoms: deterministic single photon subtraction from arbitrary light fields,” Phys. Rev. Lett. 107, 093601 (2011).
[Crossref]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref]

2010 (4)

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).
[Crossref]

K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010).
[Crossref]

C. Guerlin, E. Brion, T. Esslinger, and K. Mølmer, “Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble,” Phys. Rev. A 82, 053832 (2010).
[Crossref]

A. E. B. Nielsen and K. Mølmer, “Deterministic multimode photonic device for quantum-information processing,” Phys. Rev. A 81, 043822 (2010).
[Crossref]

2009 (2)

M. O. Scully, “Collective Lamb shift in single photon Dicke superradiance,” Phys. Rev. Lett. 102, 143601 (2009).
[Crossref]

L. H. Pedersen and K. Mølmer, “Few qubit atom-light interfaces with collective encoding,” Phys. Rev. A 79, 012320 (2009).
[Crossref]

2008 (4)

J. Bochmann, M. Mücke, G. Langfahl-Klabes, C. Erbel, B. Weber, H. P. Specht, D. L. Moehring, and G. Rempe, “Fast excitation and photon emission of a single-atom-cavity system,” Phys. Rev. Lett. 101, 223601 (2008).
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K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature 452, 67–71 (2008).
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H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
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2005 (2)

A. T. Black, J. K. Thompson, and V. Vuletić, “On-demand superradiant conversion of atomic spin gratings into single photons with high efficiency,” Phys. Rev. Lett. 95, 133601 (2005).
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A. Gilchrist, N. K. Langford, and M. A. Nielsen, “Distance measures to compare real and ideal quantum processes,” Phys. Rev. A 71, 062310 (2005).
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2004 (2)

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurášek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432, 482–486 (2004).
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L.-M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett. 92, 127902 (2004).
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2002 (3)

A. Galindo and M. A. Martín-Delgado, “Information and computation: classical and quantum aspects,” Rev. Mod. Phys. 74, 347–423 (2002).
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M. Saffman and T. G. Walker, “Creating single-atom and single-photon sources from entangled atomic ensembles,” Phys. Rev. A 66, 065403 (2002).
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R. G. Unanyan and M. Fleischhauer, “Efficient and robust entanglement generation in a many-particle system with resonant dipole-dipole interactions,” Phys. Rev. A 66, 032109 (2002).
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2001 (1)

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
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2000 (1)

W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314 (2000).
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1998 (1)

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1992 (1)

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1975 (1)

M.-D. Choi, “Completely positive linear maps on complex matrices,” Linear Algebra Appl. 10, 285–290 (1975).
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1972 (1)

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1963 (1)

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1954 (1)

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Supplementary Material (1)

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Figures (8)

Fig. 1.
Fig. 1. Schematic of the intracavity Rydberg-blocked atomic ensemble system under investigation. The entire ensemble has a size of that is on the order of a few tens of micrometers, which is compatible with the experimentally attainable Rydberg-blockade radius. Moreover, in such a configuration, the atomic ensemble well matches the cavity mode spatially, which conveniently serves the purpose of atom–photon coupling. The control laser is of frequency ωc, while the cavity resonance frequency is ωd.
Fig. 2.
Fig. 2. Schematic of the system for the study of the JCM with an intracavity Rydberg-blocked atomic ensemble, where the state initialization includes the process of feeding a nonclassical optical pulse into the cavity. The optical cavity is supposed to be single-sided, where the one end mirror is perfectly reflecting. In this particular example, the feeding and retrieving of the intracavity optical field is realized by polarization optics for the free-space optical pulse (PBS: polarizing beam splitter; QWP: quarter-wave plate). Typical parameter settings for the experimental implementation are relatively mild for the hardware nowadays. For example, the cavity finesse can be set as F5×103, the cavity decay rate can be chosen as κ2π×0.5  MHz, the cavity free spectral range can be set as FSR5  GHz, and the number of atoms in the ensemble can be chosen as 1001000. The instantiation of the Jaynes–Cummings model in such a system is relatively straightforward compared with the case of a high-finesse cavity, thanks to recent developments in single-photon pulse engineering and Rydberg atom control techniques.
Fig. 3.
Fig. 3. Numerical simulation of the quantum Rabi oscillations of the intracavity atom–photon interaction, single-photon–state case. Parameters are set as Δe=2π×200  MHz, Δr=0, Γe=2π×1  MHz, Γr=2π×0.01  MHz, κ=2π×0.5  MHz. For (a), Ω=2π×20  MHz, G=2π×10  MHz; for (b), Ω=2π×5  MHz, G=2π×2.5  MHz.
Fig. 4.
Fig. 4. Numerical simulation of the quantum Rabi oscillations of the intracavity atom–photon interaction, coherent-state version. The artificial condition of almost no cavity decay is imposed to show the quantum revival, while the atomic decays are retained in the calculations. Parameters are set as G=2π×2.5  MHz, Ω=2π×20  MHz, Δe=2π×100  MHz, Δr=0, Γe=2π×1  MHz, Γr=2π×0.01  MHz, κ=2π×104  MHz. The initial condition is |α|=4 for the optical coherent state, |α, while the cutoff is set at 50. It is averaged over 10,000 MCWF traces.
Fig. 5.
Fig. 5. Numerical simulation of the quantum Rabi oscillations of the intracavity atom–photon interaction, coherent-state version. A moderate cavity decay into the free-space environment is considered here. Parameters are set as G=2π×2.5  MHz, Ω=2π×20  MHz, Δe=2π×100  MHz, Δr=0, Γe=2π×1  MHz, Γr=2π×0.01  MHz, κ=2π×0.1  MHz. The initial condition is |α|=4 for the optical coherent state, |α, while the cutoff is set at 50. It is averaged over 10,000 MCWF traces.
Fig. 6.
Fig. 6. Outline for the basic principles of the atom–photon gate with an intracavity Rydberg-blocked atomic ensemble, where the qubit state on the atom side is abstracted into the internal states of a single atom within the ensemble. The optical cavity is supposed to be single-sided, where the one end mirror is perfectly reflecting. The frequency of the incident single-photon pulse is resonant with the optical cavity, but not necessarily so with the atomic transition |g|e. Here, the matter qubit is instantiated in the form of a single atom within the atomic ensemble. In such a configuration, a strong Rydberg blockade is presumed to take place between state |r of the qubit atom and state |r for the rest atoms of the ensemble, as a consequence of the Förster resonance structure in Eq. (11). Nevertheless, the single qubit atom does not have to be the same species as the other atoms in the ensemble [26,50]. In principle, this gate protocol does not induce mechanical forces between atoms since the underlying mechanism belongs to the category of a Rydberg-blockade gate [28].
Fig. 7.
Fig. 7. Numerical simulation of the gate’s fidelities with respect to different single-atom single-photon coupling strengths, G0. Particular parameters for this simulation include Ω=2π×100  MHz, Δe=2π×1000  MHz, Γe=2π×1  MHz, Γr=Γp=2π×0.01  MHz, κ=2π×1  MHz. While all other parameters are not changed, continuing to increase the strength of G won’t unlimitedly enhance the fidelity. This is because, as G increases beyond certain point, it will cause a power broadening effect, which harms the desired conditional phase shift. This can also be observed quantitatively from Eqs. (14) and (15).
Fig. 8.
Fig. 8. Numerical simulation of the gate’s fidelities to demonstrate its tuning range in Δe, with respect to different Rabi frequency settings of the control laser. Particular parameters for this simulation includes G0=2π×0.5  MHz, Γr=Γp=2π×0.01  MHz, κ=2π×1  MHz. The practical strategy of choosing the value of Δe will involve considerations to ensure that |Δe|Γe and |Δe|G.

Equations (20)

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Hint=n=1N(Ω2|rnen|iGn|engn|b^)+H.c.Δen=1N|enen|Δrn=1N|rnrn|+n=1Nm>nVnm|rnrn||rmrm|,
Hint=(Ω2|r˜e˜|iG|e˜gN|b^)+H.c.Δe|e˜e˜|Δr|r˜r˜|,
|e˜=G1Gn|en,|r˜=G1Gn|rn,
Heff=ΩG2Δe|r˜gN|+H.c.+G2Δe|gNgN|b^b^(ΔrΩ24Δe)|r˜r˜|,
|Ψ(t)=dωϕs(ω,t)a^s(ω)|gN,Øb,Øa+Cb(t)b^|gN,Øb,Øa+m=1NCem(t)|gN1em,Øb,Øa+m=1NCrm(t)|gN1rm,Øb,Øa,a,
Ce=G1Gn*Cen,Cr=G1Gn*Crn,
Hio=idωgs(ω)(a^s(ω)b^b^a^s(ω)).
ddtCb(t)=GCeκ2Cb(t),
ddtCe(t)=GCb(t)+iΩ*2Cr(t)+iΔeCe(t)Γe2Ce(t),
ddtCr(t)=iΩ2Ce(t)+iΔrCr(t)Γr2Cr(t),
ddtCb(t)=iG2Δe+iΓe2Cb(t)GΩ*2(Δe+iΓe2)Cr(t)κ2Cb(t),ddtCr(t)=GΩ2(Δe+iΓe2)Cb(t)i|Ω|24(Δe+iΓe2)Cr(t)+iΔrCr(t).
ddtCb,n(t)=nGCe,n1(t),
ddtCe,n1(t)=nGCb,n(t)+iΩ*2Cr,n1(t)+iΔeCe,n1(t),
ddtCr,n1(t)=iΩ2Ce,n1(t)+iΔrCr,n1(t),
Hp1=n=1N(Ω2|rnen|iGn|engn|b^)+H.cΔen=1N|enen|Δrn=1N|rnrn|+n=1NVn|rnpn||rp|+H.c.+δpn=1N|pnpn||pp|,
|Ψ(t)=dωφs(ω,t)a^s(ω)|gN,r,Øb,Øa+Cb(t)b^|gN,r,Øb,Øa+m=1NCem(t)|gN1,em,r,Øb,Øa+m=1NCrm(t)|gN1rm,r,Øb,Øa+m=1NCpm(t)|gN1pm,p,Øb,Øa.
Δac=G2Δe+δ+iΓe2,η=14|Ω|2(iΔeΓe2+iδ)2,Δdr=Δri4|Ω|2Γe2iΔeiδ,Bm=|Vm|2Γp2+iδpiδ.
R(δ)=1κ{κ2iδiΔacηm=1N|Gm|2Γr2iδ+iΔdr+Bm}1.
R(δ)=1κ{κ2iδiΔacηG2Γr2iδ+iΔdr}1.
Fz=116|2+RR|2.