aa r X i v : . [ qu a n t - ph ] O c t Phase sensitive photonic flash
X. Y. Cui, J. H. Wu, , and Z. H. Wang, , , ∗ Center for Quantum Sciences and School of Physics and Center for AdvancedOptoelectronic Functional Materials Research and Key Laboratory for UV Light-EmittingMaterials and Technology of Ministry of Education, Northeast Normal University, Changchun130024, China Beijing Computational Science Research Center, Beijing 100094, China ∗ [email protected] Abstract:
We theoretically propose a photonic flash based on a linearlycoupled cavity system. Via driving the two side cavities by external fields,it forms a cyclic energy-level diagram and therefore the phase di ff erencebetween the driving fields acts as a controller of the steady state due to thequantum interference e ff ect. In the optical trimer structure, we show that theperfect photonic flash can be realized in the situation of resonant driving.The perfect photonic flash scheme is furthermore generalized to multiplecoupled cavity system, where the cavities with odd and even number turnbright and dark alternatively. Our proposal may be applied in the designingof quantum neon and realizing a controllable photonic localization. © OCIS codes: (230.4555) Coupled resonators; (270.5580) Quantum electrodynamics;(270.5585) Quantum information and processing.
References and links
1. M. Bayindir, B. Temelkuran, and E. Ozbay, “Propagation of photons by hopping: A waveguiding mechanismthrough localized coupled cavities in three-dimensional photonic crystals”, Phys. Rev. B , R11855 (2000).2. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip”,Nature , 925 (2003).3. A. Wallra ff , D. I. Schuster, A. Blais, L. Frunzio, R.- S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J.Schoelkopf, “Circuit quantum electrodynamics: Coherent coupling of a single photon to a Cooper pair box”,Nature
162 (2004).4. A. Blais, R.-Shou Huang, A. Wallra ff , S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum electrodynamics forsuperconducting electrical circuits: An architecture for quantum computation”, Phys. Rev. A , 062320 (2004).5. L. Zhou, Z. R. Gong, Y.-x. Liu, C. P. Sun, and F. Nori, “Controllable scattering of a single photon inside aone-dimensional resonator waveguide”, Phys. Rev. Lett. , 100501 (2008).6. L. Zhou, L.-P. Yang, Y. Li, and C. P. Sun, “Quantum routing of single photons with a cyclic three-level system”,Phys. Rev. Lett. , 103604 (2013).7. Z. H. Wang, L. Zhou, Y. Li, and C. P. Sun, “Controllable single-photon frequency converter via a one-dimensionalwaveguide”, Phys. Rev. A , 053813 (2014).8. A. Zheng, G. Zhang, L. Gui, and J. Liu, “Single-photon frequency conversion and multi-mode entanglement viaconstructive interference on Sagnac loop”, Laser Phys. , 065201 (2015).9. W.-B. Yan, J.-F. Huang, and H. Fan, “Tunable single-photon frequency conversion in a Sagnac interferometer”,Sci. Rep. , 3555 (2013).10. F. Nissen, S. Schmidt, M. Biondi, G. Blatter, H. E. Tureci, and J. Keeling, “Nonequilibrium dynamics of coupledqubit-cavity arrays”, Phys. Rev. Lett. , 233603 (2012).11. A. Le Boite, G. Orso, and C. Ciuti, “Steady-state phases and tunneling-induced instabilities in the driven dissi-pative Bose-Hubbard model”, Phys. Rev. Lett. , 233601 (2013).12. J. Jin, D. Rossini, R. Fazio, M. Leib, and M. J. Hartmann, “Photon solid phases in driven arrays of nonlinearlycoupled cavities”, Phys. Rev. Lett. , 163605 (2013).13. J. Raftery, D. Sadri, S. Schmidt, H. E. Tureci, and A. A. Houck, “Observation of a dissipation-induced classicalto quantum transition”, Phys. Rev. X , 031043 (2014).4. J. Jin, A. Biella, O. Viyuela, L. Mazza, J. Keeling, R. Fazio, and D. Rossini, “Cluster mean-field approach to thesteady-state phase diagram of dissipative spin systems”, Phys. Rev. X , 031011 (2016).15. J. J. Mendoza-Arenas, S. R. Clark, S. Felicetti, G. Romero, E. Solano, D. G. Angelakis, and D. Jaksch, “Beyondmean-field bistability in driven-dissipative lattices: Bunching-antibunching transition and quantum simulation”,Phys. Rev. A , 023821 (2016).16. C. Noh and D. G Angelakis, “Quantum simulations and many-body physics with light”, Rep. Prog. Phys. ,016401 (2017).17. S. Schmidt and J. Koch, “Circuit QED lattices: towards quantum simulation with superconducting circuits”, Ann.Phys. (Amsterdam) , 395 (2013).18. Z. H. Wang, X.-W. Xu, and Y. Li, “Partially dark optical molecule via phase control”, Phys. Rev. A , 013815(2017).19. Y.-X. Liu, J. Q. You, L. F. Wei, C. P. Sun, and F. Nori, “Optical selection rules and phase-dependent adiabaticstate control in a superconducting quantum circuit”, Phys. Rev. Lett. , 087001 (2005).20. W. Z. Jia and L. F. Wei, “Gains without inversion in quantum systems with broken parities”, Phys. Rev. A ,013808 (2010).21. M. Shapiro, E. Frishman, and P. Brumer, “Coherently controlled asymmetric synthesis with achiral light”, Phys.Rev. Lett. , 1669 (2000).22. P. Kral and M. Shapiro, “Cyclic population transfer in quantum systems with broken symmetry”, Phys. Rev. Lett. , 183002 (2001).23. P. Kral, I. Thanopulos, M. Shapiro, and D. Cohen, “Two-step enantio-selective optical switch”, Phys. Rev. Lett. , 033001 (2003).24. Y. Li, C. Bruder, and C. P. Sun, “Generalized Stern-Gerlach e ff ect for chiral molecules”, Phys. Rev. Lett. ,130403 (2007).25. J. Tang, W. Geng, and X. Xu, “Quantum interference induced photon blockade in a coupled single quantumdot-cavity system”, Sci. Rep. , 9252 (2015).26. Y. Liu and D. L. Zhou, “Quantum state transfer along a ring with time-reversal asymmetry”, Phys. Rev. A ,052318 (2015).27. W. Z. Jia, L. F. Wei, Y. Li, and Y.-x. Liu, “Phase-dependent optical response properties in an optomechanicalsystem by coherently driving the mechanical resonator’, Phys. Rev. A , 043843 (2015).28. X. W. Xu and Y. J. Li, “Antibunching photons in a cavity coupled to an optomechanical system”, J. Phys. B ,035502 (2013).29. P. W. Anderson, “Absence of di ff usion in certain random lattices”, Phys. Rev. , 1492 (1958).30. A. Pal and D. A. Huse, “Many-body localization phase transition”, Phys. Rev. B , 174411 (2010).
1. Introduction
Coherent photonic control and designing photonic devices are of potential application in quan-tum information processing, and the coupled cavity array system provides an ideal platform forachieving such tasks.In the coupled cavity array system, which can be realized by the photonic crystal [1], toroidmicocavity [2] as well as superconductive transmission line [3,4], the coherent photonic controlhas invoked a lot of attentions. In the ideal situation, where the decay of the cavities is neglected,kinds of single-photon devices have been theoretically proposed within the scattering frame,such as quantum transistor [5], quantum router [6] and quantum frequency converter [7–9]. Onthe other hand, in the coupled cavity array with loss, it has been shown that the non-equilibriumquantum phase transition, which is related to the steady state, may occur in some nonlinear sys-tem when the driving and dissipation are both present [10–15]. Therefore, it supplies a photonicplatform to simulate many body phenomenon in condensed matter physics [16, 17]. Naturally,the steady state in the coupled cavity array with only linear interaction deserves more explo-ration.In this paper, by investigating the photonic distribution of steady state in a linearly coupledcavity array system, we propose a scheme to realize photonic flash. We here use the phrase“photonic flash” to refer that when some of cavities in the coupled cavity array are completelydark, the other ones are completely bright, and the bright and dark cavities can be exchangedby tunning the system parameters appropriately. In our scheme, the two side cavities are drivenby a pair of external fields, we will show that the phase di ff erence between the driving fields ig. 1. The scheme of an optical trimer for photonic flash, where the two side cavities aredriven by classical fields. acts as an ideal controller for the steady state, which stems from the quantum interference e ff ectamong the multiple transition channels. As a simple example, we sketch the realization of darkcavity in an optical trimer, and show that we can construct a perfect photonic flash under thesituations (i) the central cavity of the trimer is lossless and (ii) the two side cavities are drivenresonantly. Generalizing the scheme to an array of multiple coupled cavities, we show thatthe cavities with odd and even number will be bright and dark alternatively by adjusting thephase di ff erence. Therefore, we design a scalable photonic flash based on linear coupled cavityarray system. Instead of controlling each cavity individually, we here only drive the two sidecavities, regardless of the total number of the cavities, and therefore is beneficial to avoidingthe di ffi culties of addressing. As the potential application in the designing of photonic deviceand quantum simulation based on our proposal, we also give a simple remark on the realizingof quantum neon and controllable photonic localization.The rest of the paper is organized as follows. In Sec. 2, we present the model of an opticaltrimer and the steady state values. In Sec. 3, we discuss the condition to realize perfect photonicflash in the trimer structure and generalize it to multi-coupled-cavity system in Sec. 4. At last,we discuss the e ff ect of quantum noise and give a brief remark and conclusion in Sec. 5.
2. Model of optical trimer
The optical trimer system in our consideration is sketched in Fig. 1, in which the cavity 1 , ω d . In the rotating frame with respect to the frequency of the driving fields, the Hamiltoniancan be written as (here and after ¯ h = H = X i = ∆ i a † i a i + J ( a † a + a † a + a † a + a † a ) + λ ( a e i ϕ + a † e − i ϕ ) + λ ( a + a † ) , (1)where ∆ i = ω i − ω d ( i = , ,
3) is the detuning between the i th cavity and the driving fields.Here, ω i is the resonant frequency of cavity i , which is described by the annihilation operator a i . J is nearest inter-cavity coupling strength. λ and λ are the driving strengths to cavity 1and 3, respectively. ϕ is the phase di ff erence between the two driving fields, which will play animportant role in the controlling of photons. Without loss of generality, we assume that all theparameters are real.Assuming the decay rate of the i th cavity is γ i ( > A = M A + B , (2)here A = ( a , a , a ) T , B = ( − i λ e − i ϕ , , − i λ ) T , and the matrix M is M = − i ∆ − γ / − iJ − iJ − i ∆ − γ / − iJ − iJ − i ∆ − γ / . (3)Then, the steady-state values of the system are immediately given by h a i = iJ [ λ − λ e − i ϕ (1 + K )]det(M) , (4a) h a i = − J [ λ e − i ϕ ( i ∆ + γ / + λ ( i ∆ + γ / , (4b) h a i = iJ [ λ e − i ϕ − λ (1 + K )]det(M) . (4c)where K m : = ( i ∆ + γ / i ∆ m + γ m / / J , m = ,
3. Therefore, the steady state property is sen-sitively dependent on the phase di ff erence ϕ between the two driving fields. This phase depen-dence mechanism is similar to those in optical molecule (dimer) system [18], superconductingartificial [19,20], chiral molecule [21–24], cavity-QED system [25,26] and cavity optomechan-ical system [27, 28], where the coupling and driving contribute a cyclic energy-level structure.
3. Photonic flash in the trimer
As discussed above, the phase di ff erence is a potential controller for the photonic state in anoptical trimer, which supplies us a convenient way to design the coherent optical device, suchas photonic flash, utilizing the interference e ff ect. Here, we use the phrase “photonic flash”to mean that, by adjusting only the phase di ff erence between the driving fields, we will makeone or few of the cavities reaches their vacuum steady state, while the intensity in other cav-ities reach their maximum values. In other words, we will realize the bright and dark cavitiessimultaneously, and they can be controllably swapped.To be explicit, we consider a simple scheme in which ∆ = ∆ = ∆ = ∆ , λ = λ = λ = λ, γ = γ = γ, γ = γ ′ . According to the formula in Eqs. (4), the condition for h a i = ϕ = J + γγ ′ − ∆ J , sin ϕ = ∆ γ + ∆ γ ′ J , (5)and the condition for h a i = ϕ = J + γγ ′ − ∆ J , sin ϕ = − ∆ γ + ∆ γ ′ J . (6)Obviously, the above two equations imply (cid:16) J + γγ ′ − ∆ (cid:17) + (cid:0) ∆ γ + ∆ γ ′ (cid:1) = J . (7)Furthermore, in the case of ϕ = π , we will obtain that h a i = ff erence ϕ . Under the condition ϕ/ π |h a i| |h a i| |h a i| Fig. 2. The steady state values of the system as a function of the phase di ff erence ϕ betweentwo driving fields in the non-resonant driving situation. The parameters are set as J = γ,λ = γ , ∆ = . γ,γ ′ = . γ . Under these parameters, the condition in Eq. (7) is satisfied. in Eq. (7), the photonic intensity in each cavity can be expressed in a compact form as |h a i| = δ ( ∆ + γ /
4) [1 + cos( ϕ + θ )] , (8a) |h a i| = δ J (1 + cos ϕ ) , (8b) |h a i| = δ ( ∆ + γ /
4) [1 + cos( ϕ − θ )] , (8c)where δ : = p λ J / (4 J − ∆ + γγ ′ /
2) andcos θ = − J + γγ ′ − ∆ J , sin θ = ∆ ( γ + γ ′ )2 J . (9)The phase dependence of steady state values in the case of ∆ = . γ is shown in Fig. 2,where we plot |h a i i| ( i = , ,
3) as a function of ϕ . It shows that, |h a i| = ϕ = π , and |h a i| = ϕ ≈ . . π . In other words, we can realize the partially dark cavities inthe trimer system, which is similar to that in optical molecule or dimer system [18]. However,as shown in Fig. 2, when any of the cavities is dark with zero average photons in the steadystate, the photonic intensity in the other two cavities can not achieve their maximum values.That is, we can not construct a perfect photonic flash. This defect can be overcome by drivingthe system resonantly, that is ∆ =
0, in which situation we need a lossless condition γ ′ = θ = π and the average photon number become |h a i| = |h a i| = λ (1 − cos ϕ ) /γ and |h a i| = λ (1 + cos ϕ ) / (2 J ). In Fig. 3, we plot the curve of the steady values for ∆ = ϕ = π , we will have |h a i| =
0, and |h a i| = |h a i| , achieving their maximumvalues. On contrary, by adjusting ϕ = π ), the two side cavities in the trimer becomecompletely dark, that is |h a i| = |h a i| =
0, and the central cavity becomes bright. In Fig. 3,we have chosen the inter-coupling strength as J = . γ , so that we will reach that the maximumvalues of |h a i| , |h a i| and |h a i| are equal to each other. In such a way, we have constructeda perfect photonic flash only by tunning the phase di ff erence in an optical trimer system. ϕ/ π |h a , i| |h a i| Fig. 3. The steady state values of the system as a function of the phase di ff erence ϕ betweentwo driving fields in the resonant driving situation. The parameters are set as J = . γ,λ = γ , ∆ = ,γ ′ =
0. Under these parameters, the condition in Eq. (7) is satisfied.
4. Generalization to coupled cavity array
The scheme of phase dependent photonic flash can also be naturally generalized to the system of N ( >
3) linearly coupled cavities. By driving the 1st and N th cavities classically, the Hamiltonianof the system reads (in the rotating frame) H = N X i = ∆ i a † i a i + J N − X i = ( a † i a i + + h . c . ) + λ ( a e i ϕ + a N + h . c . ) . (10)Motivated by the realization of perfect photonic flash in optical trimer structure, we here as-sume a resonant driving situation, and that all of the middle cavities are lossless by setting theparameters as ∆ i = γ i = γ ( δ i , + δ i , N ) , ( γ > , i = , , · · · , N ). The Heisenberg-Langevinequation (neglecting the quantum noise) then reads − γ a − iJa = i λ e − i ϕ , (11a) a i + a i + = , ( i = , , ... N − , (11b) − γ a N − iJa N − = i λ. (11c)Firstly, let us consider the situation for odd N by assuming N = m +
1, where m is an integer.The steady state values can be obtained from Eqs. (11) as |h a j + i| = |h a i| = λ γ [1 + ( − m cos ϕ ] , (12) |h a j i| = |h a i| = λ J [1 − ( − m cos ϕ ] , (13)for j = , , · · · m . It implies that, when cos ϕ = − ( − m , all of the odd number cavities will becompletely dark with zero intensity. Meanwhile, the photonic intensity in the even number cav-ities will achieve their maximums. The opposite event will occur as long as the phase di ff erencesatisfies cos ϕ = ( − m , for which the even number cavities become dark and the odd ones be-come bright. In Fig. 4(a), we illustrate the realization of photonic flash in an array of 5 ( m = ig. 4. The scheme of phase dependent photonic flash based on coupled cavity array system.We take (a) N = N = by the grey cavities) by tunning the phase di ff erence such that ϕ =
0, and they will turn bright(shown by the yellow cavities) when ϕ = π . At the same time, the odd number cavities willexperience an exchange from bright ones to dark ones as we adjust the phase di ff erence from ϕ = ϕ = π .Secondly, let us move to the situation of even cavities, that is N = m , where m is an integer.A simple calculation shows that the perfect photonic flash can be realized under the parameter γ = J and the steady state values yield |h a j − i| = |h a i| = λ [1 + ( − m sin ϕ ]2 J , (14) |h a j i| = |h a i| = λ [1 − ( − m sin ϕ ]2 J , (15)where j = , , · · · , m . Taking N = m =
3) as an example, we illustrate the realization of perfectphotonic flash in Fig. 4(b). As shown in the figure, the odd (even) number cavities and even(odd) number cavities become bright (dark) alternatively when ϕ experiences a change from3 π/ π/
5. Remark and Conclusion
In this paper, we have investigated how to coherently control the steady state and realize thephotonic flash in a coupled cavity array system. All of our discussions are based on the semi-classical approximation, where the decay of the side cavities are taken into consideration, butthe e ff ect of the noise is neglected. Here, the decay can be induced by for example the radiativeloss and imperfectness of cavities and the noise mainly comes from the quantum fluctuationof the electromagnetic field in the surrounding environment. To go beyond the semi-classicalapproximation, the noise contribution to the photonic intensity in an optical trimer structure isobtained as |h a i i noi | = |h a i i| + X j = n th j Z t d τ | D i j ( τ − t ) | . (16)for i = , , D ( t ) = exp( Mt ). Here |h a i i noi | is the photon number by taking the quantumnoise into account, M and |h a i i| are given in Eq. (3) and Eqs. (8), respectively. n th j is thehermal photon number in the reservoir contact with the j th cavity. Therefore the quantum noiseat zero temperature ( n th j =
0) will make no contribution to the function of the photonic flash.Usually, the optical cavities are characterized by their high eigen-frequencies and the noise canbe reasonably regarded as at zero temperature. Therefore, we believe that the semi-classicalapproximation in our discussion will give a logical prediction.In conclusion, we have theoretically proposed a scheme for photonic flash based on lin-early coupled cavity system by only tuning the phase di ff erence between the two driving fields.We firstly demonstrate how to realize a partially dark optical trimer and then design a pho-tonic flash, in which the side cavities are driven resonantly. In such a scheme, the side cavitiesachieve their vacuum steady state when the central one reaches a maximum photonic intensity,and vice versa. We also generalize the proposal of photonic flash to the system which is com-posed by multiple coupled cavities, and show that the flash occurs between the odd and evennumber cavities. In our scheme, the middle cavities are required to be lossless, which can notbe achieved strictly. However, within the current technologies, the decay rate can be negligiblysmall and satisfies our condition approximately.At last, we claim that our proposal in this paper maybe of potential application in the quan-tum device designing and quantum simulation. On one hand, in our current scheme, all of thecavities carry a same frequency, and a natural generalization is to design quantum neon. That is,the resonant frequencies of the cavities are di ff erent and so we can control the intensity of lightwith di ff erent “color” by adjusting the phase. On the other hand, since the cavities in our systemcan be dark or bright by tunning the phase di ff erence, we here actually realize a controllablephotonic localization phenomenon. Traditionally, the localization is observed in the disorderednonlinear system, such as Anderson localization [29] and many-body localization [30]. Wehope that our scheme in linear ordered cavity array system will be of potential application. Acknowledgments
We thank D. L. Zhou and Y. X. Liu for their helpful discussions.