Abstract

We describe an approach to optical non-reciprocity that exploits the local helicity of evanescent electric fields in axisymmetric resonators. By interfacing an optical cavity to helicity-sensitive transitions, such as Zeeman levels in a quantum dot, light transmission through a waveguide becomes direction-dependent when the state degeneracy is lifted. Using a linearized quantum master equation, we analyze the configurations that exhibit non-reciprocity, and we show that reasonable parameters from existing cavity QED experiments are sufficient to demonstrate a coherent non-reciprocal optical isolator operating at the level of a single photon.

© 2014 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  40. A. Kuther, M. Bayer, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, and J. P. Reithmaier, “Zeeman splitting of excitons and biexcitons in single In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B58, 7508– 7511 (1998).
    [CrossRef]
  41. R. Beaulac, P. I. Archer, S. T. Ochsenbein, and D. R. Gamelin, “Mn2+-doped CdSe quantum dots: New inorganic materials for spin-electronics and spin-photonics,” Adv. Func. Mater.18, 3873–3891 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]

2014

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Physics10, 394–398 (2014).
[CrossRef]

2013

C. Junge, D. O’Shea, J. Volz, and A. Rauschenbeutel, “Strong coupling between single atoms and nontransversal photons,” Phys. Rev. Lett.110, 213604 (2013).
[CrossRef] [PubMed]

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nat. Photonics7, 579–582 (2013).
[CrossRef]

2012

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on ‘nonreciprocal light propagation in a silicon photonic circuit’,” Science335, 38(2012).
[CrossRef]

C. Lacroûte, K. S. Choi, A. Goban, D. J. Alton, D. Ding, N. P. Stern, and H. J. Kimble, “A state-insensitive, compensated nanofiber trap,” New J. Phys.14, 023056 (2012).
[CrossRef]

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroûte, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett.109, 033603 (2012).
[CrossRef] [PubMed]

A. Pandey, S. Brovelli, R. Viswanatha, L. Li, J. M. Pietryga, V. I. Klimov, and S. Crooker, “Long-lived photoinduced magnetization in copper-doped ZnSe-CdSe core-shell nanocrystals,” Nat. Nano.7, 792–797 (2012).
[CrossRef]

D. Kilper, K. Guan, K. Hinton, and R. Ayre, “Energy challenges in current and future optical transmission networks,” Proceedings of the IEEE100, 1168–1187 (2012).
[CrossRef]

P. J. Shadbolt, M. R. Verde, A. Peruzzo, A. Politi, A. Laing, M. Lobino, J. C. F. Matthews, M. G. Thompson, and J. L. O’Brien, “Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit,” Nat. Photonics6, 45–49 (2012).
[CrossRef]

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science335, 447–450 (2012).
[CrossRef]

M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express20, 7672–7684 (2012).
[CrossRef] [PubMed]

2011

M.-C. Tien, J. F. Bauters, M. J. R. Heck, D. T. Spencer, D. J. Blumenthal, and J. E. Bowers, “Ultra-high quality factor planar Si3N4 ring resonators on Si substrates,” Opt. Express19, 13551–13556 (2011).
[CrossRef] [PubMed]

C. J. Trowbridge, B. M. Norman, J. Stephens, A. C. Gossard, D. D. Awschalom, and V. Sih, “Electron spin polarization-based integrated photonic devices,” Opt. Express19, 14845–14851 (2011).
[CrossRef] [PubMed]

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys.7, 907–912 (2011).
[CrossRef]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science333, 729–733 (2011).
[CrossRef] [PubMed]

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics5, 758–762 (2011).
[CrossRef]

Y. Shen, M. Bradford, and J. T. Shen, “Single-photon diode by exploiting the photon polarization in a waveguide,” Phys. Rev. Lett.107, 173902 (2011).
[CrossRef] [PubMed]

X. W. Mi, J. X. Bai, D. J. Li, and H. P. Zhao, “Coupling to a microdisk cavity containing a three-level quantum-dot with two orthogonal modes,” Opt. Commun.284, 2937–2942 (2011).
[CrossRef]

D. J. Alton, N. P. Stern, T. Aoki, H. Lee, E. Ostby, K. J. Vahala, and H. J. Kimble, “Strong interactions of single atoms and photons near a dielectric boundary,” Nat. Phys.7, 159–165 (2011).
[CrossRef]

2010

J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A82, 043811 (2010).
[CrossRef]

2009

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics3, 687–695 (2009).
[CrossRef]

R. Beaulac, L. Schneider, P. I. Archer, G. Bacher, and D. R. Gamelin, “Light-induced spontaneous magnetization in doped colloidal quantum dots,” Science325, 973–976 (2009).
[CrossRef] [PubMed]

H. Htoon, S. A. Crooker, M. Furis, S. Jeong, A. L. Efros, and V. I. Klimov, “Anomalous circular polarization of photoluminescence spectra of individual CdSe nanocrystals in an applied magnetic field,” Phys. Rev. Lett.102, 017402 (2009).
[CrossRef] [PubMed]

J.-T. Shen and S. Fan, “Theory of single-photon transport in a single-mode waveguide. ii. coupling to a whispering-gallery resonator containing a two-level atom,” Phys. Rev. A79, 023838 (2009).
[CrossRef]

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett.102, 213903 (2009).
[CrossRef] [PubMed]

A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science325, 1221 (2009).
[CrossRef]

A. Gondarenko, J. S. Levy, and M. Lipson, “High confinement micron-scale silicon nitride high Q ring resonator,” Opt. Express17, 11366–11370 (2009).
[CrossRef] [PubMed]

2008

R. Beaulac, P. I. Archer, S. T. Ochsenbein, and D. R. Gamelin, “Mn2+-doped CdSe quantum dots: New inorganic materials for spin-electronics and spin-photonics,” Adv. Func. Mater.18, 3873–3891 (2008).
[CrossRef]

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science320, 646–649 (2008).
[CrossRef] [PubMed]

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science319, 1062–1065 (2008).
[CrossRef] [PubMed]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett.100, 013905 (2008).
[CrossRef] [PubMed]

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett.100, 013904 (2008).
[CrossRef] [PubMed]

2007

K. Srinivasan and O. Painter, “Linear and nonlinear optical spectroscopy of a strongly coupled microdisk– quantum dot system,” Nature450, 862–865 (2007).
[CrossRef] [PubMed]

K. Srinivasan and O. Painter, “Mode coupling and cavity–quantum-dot interactions in a fiber-coupled microdisk cavity,” Phys. Rev. A75, 023814 (2007).
[CrossRef]

2006

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature443, 671–674 (2006).
[CrossRef] [PubMed]

N. Le Thomas, U. Woggon, O. Schöps, M. V. Artemyev, M. Kazes, and U. Banin, “Cavity QED with semiconductor nanocrystals,” Nano Letters6, 557–561 (2006).
[CrossRef] [PubMed]

2004

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys.67, 717 (2004).
[CrossRef]

2003

1998

A. Kuther, M. Bayer, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, and J. P. Reithmaier, “Zeeman splitting of excitons and biexcitons in single In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B58, 7508– 7511 (1998).
[CrossRef]

1985

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A31, 3761 (1985).
[CrossRef] [PubMed]

Alton, D. J.

C. Lacroûte, K. S. Choi, A. Goban, D. J. Alton, D. Ding, N. P. Stern, and H. J. Kimble, “A state-insensitive, compensated nanofiber trap,” New J. Phys.14, 023056 (2012).
[CrossRef]

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroûte, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett.109, 033603 (2012).
[CrossRef] [PubMed]

D. J. Alton, N. P. Stern, T. Aoki, H. Lee, E. Ostby, K. J. Vahala, and H. J. Kimble, “Strong interactions of single atoms and photons near a dielectric boundary,” Nat. Phys.7, 159–165 (2011).
[CrossRef]

Aoki, T.

D. J. Alton, N. P. Stern, T. Aoki, H. Lee, E. Ostby, K. J. Vahala, and H. J. Kimble, “Strong interactions of single atoms and photons near a dielectric boundary,” Nat. Phys.7, 159–165 (2011).
[CrossRef]

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science319, 1062–1065 (2008).
[CrossRef] [PubMed]

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature443, 671–674 (2006).
[CrossRef] [PubMed]

Archer, P. I.

R. Beaulac, L. Schneider, P. I. Archer, G. Bacher, and D. R. Gamelin, “Light-induced spontaneous magnetization in doped colloidal quantum dots,” Science325, 973–976 (2009).
[CrossRef] [PubMed]

R. Beaulac, P. I. Archer, S. T. Ochsenbein, and D. R. Gamelin, “Mn2+-doped CdSe quantum dots: New inorganic materials for spin-electronics and spin-photonics,” Adv. Func. Mater.18, 3873–3891 (2008).
[CrossRef]

Artemyev, M. V.

N. Le Thomas, U. Woggon, O. Schöps, M. V. Artemyev, M. Kazes, and U. Banin, “Cavity QED with semiconductor nanocrystals,” Nano Letters6, 557–561 (2006).
[CrossRef] [PubMed]

Awschalom, D. D.

Ayache, M.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science333, 729–733 (2011).
[CrossRef] [PubMed]

Ayre, R.

D. Kilper, K. Guan, K. Hinton, and R. Ayre, “Energy challenges in current and future optical transmission networks,” Proceedings of the IEEE100, 1168–1187 (2012).
[CrossRef]

Bacher, G.

R. Beaulac, L. Schneider, P. I. Archer, G. Bacher, and D. R. Gamelin, “Light-induced spontaneous magnetization in doped colloidal quantum dots,” Science325, 973–976 (2009).
[CrossRef] [PubMed]

Baets, R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nat. Photonics7, 579–582 (2013).
[CrossRef]

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on ‘nonreciprocal light propagation in a silicon photonic circuit’,” Science335, 38(2012).
[CrossRef]

Bai, J. X.

X. W. Mi, J. X. Bai, D. J. Li, and H. P. Zhao, “Coupling to a microdisk cavity containing a three-level quantum-dot with two orthogonal modes,” Opt. Commun.284, 2937–2942 (2011).
[CrossRef]

Banin, U.

N. Le Thomas, U. Woggon, O. Schöps, M. V. Artemyev, M. Kazes, and U. Banin, “Cavity QED with semiconductor nanocrystals,” Nano Letters6, 557–561 (2006).
[CrossRef] [PubMed]

Bauters, J. F.

Bayer, M.

A. Kuther, M. Bayer, A. Forchel, A. Gorbunov, V. B. Timofeev, F. Schäfer, and J. P. Reithmaier, “Zeeman splitting of excitons and biexcitons in single In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B58, 7508– 7511 (1998).
[CrossRef]

Beaulac, R.

R. Beaulac, L. Schneider, P. I. Archer, G. Bacher, and D. R. Gamelin, “Light-induced spontaneous magnetization in doped colloidal quantum dots,” Science325, 973–976 (2009).
[CrossRef] [PubMed]

R. Beaulac, P. I. Archer, S. T. Ochsenbein, and D. R. Gamelin, “Mn2+-doped CdSe quantum dots: New inorganic materials for spin-electronics and spin-photonics,” Adv. Func. Mater.18, 3873–3891 (2008).
[CrossRef]

Bender, C. M.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Physics10, 394–398 (2014).
[CrossRef]

Bi, L.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics5, 758–762 (2011).
[CrossRef]

Blumenthal, D. J.

Bowen, W. P.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature443, 671–674 (2006).
[CrossRef] [PubMed]

Bowers, J. E.

Bradford, M.

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Figures (7)

Fig. 1
Fig. 1

(a) The proposed non-reciprocal device acting as an optical diode. (b) Forward (blue) and backward (red) transmission spectra for a nonzero QD excited state splitting. Coupling between the counter-propagating resonator modes and the helicity-sensitive Zeeman-split excited states of the QD results in directional asymmetry of the spectra. At a certain frequency (shown by the dashed line) the system exhibits a high degree of contrast Tf/Tb.

Fig. 2
Fig. 2

Cross-sectional plots of normalized |E| and P for the M = 129 CW fundamental (ab) quasi-TE and (c–d) quasi-TM modes of a silicon nitride microring of outer radius 20 μm. Arrows indicate the direction of the E field in the ρ-z plane. The frequency of this particular mode is ωC/2π = 194.0 THz, although the general features are independent of M, geometry, and frequency. The region of the evanescent field directly (a–b) on top or (c–d) on the side of the cavity has a value of P near unity and a large value of |E|. Finite element calculations were performed with COMSOL Multiphysics [28].

Fig. 3
Fig. 3

(a) Schematic of our non-reciprocal optical device, consisting of a waveguide evanescently coupled to a WGM cavity interacting with a QD at (ρ′, ϕ′, z′). κex is the coupling from the waveguide into the cavity and vice-versa, κi is the decay rate of the cavity modes, and h is the backscattering in the cavity. Tf and Tb are the transmission for forward and backward propagating light in the waveguide respectfully. (b) Energy levels of the quantum dot. The two excitonic excited states couple to electric fields of opposite helicity.

Fig. 4
Fig. 4

Exciton-polariton energy eigenvalues λE of as a function of (a) the splitting δ12 in the ideal case, and (b) the helicity p. Dashed lines indicate the conditions for the spectra in Figs. 5(a)–5(b) and blue (red) indicates the forward (backward) sub-system. Other parameters are (g0, κi, κex) = (20, 3, 5)γ. The three eigenvalues for p = 0 are characteristic of standard cavity QED with axisymetric resonators [24, 25, 32].

Fig. 5
Fig. 5

Forward (blue) and backward (red) transmission spectra in (a) the ideal case with zero excited-state splitting, (b) the ideal case with non-zero splitting, and (c–d) the non-ideal case with non-zero splitting. Other parameters are (g0, κi, κex) = (20, 3, 5)γ, ϑ = π/4.

Fig. 6
Fig. 6

(a) The forward transmission with δ12 and ΔC set by Eqs. (18)(19). (b) The splitting δ12 set by Eq. (18) for several values of g0. The dashed line shows the optimal value of κex to maximize Tf for g0 = 20γ. Other parameters are (h, κi) = (0, 5)γ, p = 1, and ϑ = π/4.

Fig. 7
Fig. 7

Contour plots of (a) the isolation contrast Tf/Tb and (b) Tf for variable κex and δ12 in the non-ideal case of h = 20γ, p = 0.8 and ΔC set to the cavity-like dip in the Tb spectrum. The dashed line indicates where Tb = 0 may occur and the cross indicates the parameter location of optimal contrast. Other parameters are (g0, κi) = (20, 5)γ and ϑ = π/4.

Equations (19)

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E CW ( r ) = E M ( ρ , z ) e iM ϕ = ( E ρ , i E ϕ , E z ) e iM ϕ
E CCW ( r ) = E M ( ρ , z ) e iM ϕ = ( E ρ , i E ϕ , E z ) e iM ϕ
e ^ ± ( ρ , z ) = ( e ^ ± i ϕ ^ ) / 2 1 / 2
P ( ρ , z ) = ( | E + | 2 | E | 2 ) / | E | 2
E CW , ± ( r ) = | E M ( ρ , z ) | e iM ϕ [ ( 1 ± p ( ρ , z ) ) / 2 ] 1 / 2
E CCW , ± ( r ) = | E M ( ρ , z ) | e iM ϕ [ ( 1 p ( ρ , z ) ) / 2 ] 1 / 2
C = ω C ( a a + b b ) + h ( a b + b a ) + p * e i ω p t o + p e i ω p t o
C = Δ C ( a a + b b ) + h ( a b + b a ) + p * o + p o
QD = ( ω QD + δ 12 / 2 ) σ 1 + σ 1 + ( ω QD δ 12 / 2 ) σ 2 + σ 2
int = d ^ E ^
E ^ ( ρ , ϕ , z ) = E CW , + e ^ + a + E CW , e ^ a + E CCW , + e ^ + b + E CCW , e ^ b + c . c .
g ( r ) = d | E M ( ρ , z ) | e iM ϕ = g 0 ( ρ , z ) e i ϑ
int = ( g + a σ 1 + + g + * σ 1 a ) + ( g a σ 2 + + g * σ 2 a ) + ( g * b σ 1 + + g σ 1 b ) + ( g + * b σ 2 + + g + σ 2 b )
ρ ˙ = i [ , ρ ] + 2 κ L ( a ) ρ + 2 κ L ( b ) ρ + γ L ( σ 1 ) ρ + γ L ( σ 2 ) ρ
T f = | i + 2 κ ex a / p | 2
R f = | 2 κ ex b / p | 2
T f , b = | 1 2 κ ex [ γ / 2 + i ( Δ C ± δ 12 / 2 ) ] g 0 2 + [ γ / 2 + i ( Δ C ± δ 12 / 2 ) ] ( κ ex + κ i + i Δ C ) | 2
δ 12 = [ γ 2 ( κ ex κ i ) ] ( 2 g 0 2 γ ( κ ex κ i ) 1 ) 1 / 2
Δ C = ( κ ex κ i ) ( 2 g 0 2 γ ( κ ex κ i ) 1 ) 1 / 2

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