aa r X i v : . [ qu a n t - ph ] O c t Optical quantum swapping in a coherent atomic medium
Aur´elien Dantan
QUANTOP, Danish National Research Foundation Center for Quantum Optics,Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark
We propose to realize a passive optical quantum swapping device which allows for the exchange of thequantum fluctuations of two bright optical fields interacting with a coherent atomic medium in an optical cavity.The device is based on a quantum interference process between the fields within the cavity bandwidth arisingfrom coherent population trapping in the atomic medium.
PACS numbers: 42.50.Ex,42.50.Lc,42.50.Gy
Manipulating the quantum state of bright optical fields is atthe core of quantum information processing in the continuousvariable regime [1–3]. However, even basic linear processingtasks, such as the quantum state swapping between two dif-ferent frequency fields, can be challenging to achieve in prac-tice. Unconditional exchange of quantum fluctuations can beachieved for instance using complex teleportation and entan-glement swapping protocols [4–7], which typically require en-tanglement and / or measurements combined with active feed-back [8, 9]. We propose here to realize a passive quantumswapping device which allows for an e ffi cient exchange of thequantum fluctuations of two optical beams interacting in anoptical cavity with a coherent atomic medium consisting ofthree-level Λ atoms. In a coherent population trapping (CPT)situation, when the two fields are resonant with the ground-to-excited state transitions and strongly drive the atoms into a co-herent superposition of the two ground-states, the medium be-comes transparent for the fields [10]. Like in Electromagneti-cally Induced Transparency (EIT) this behavior occurs withina certain frequency window around two-photon resonance de-fined by the e ff ective cavity linewidth κ CPT , which, for a suf-ficiently high e ff ective optical depth of the medium, can bemuch narrower than that of the bare cavity κ [11–16]. Weshow that, like the field classical mean-values, the quantumfluctuations are also preserved within this transparency win-dow. However, in the frequency range κ CPT < ω < κ , where ω is the frequency of the sidebands considered, both fieldsare shown to exchange their respective fluctuations, thus real-izing a quantum swapping operation. This exchange arisesfrom the CPT-induced quantum interferences which a ff ectin a di ff erent way the dark and bright field mode combina-tions which are uncoupled and coupled, respectively, with theatomic medium. We show that, for a wide range of param-eters, the system can act as a lossless, frequency-dependentphase-plate for the field sidebands and achieve e ffi cient quan-tum state swapping.Quantum interference e ff ects due to coherent populationtrapping or electromagnetically induced transparency, in par-ticular the strong dispersion and low absorption that can beexperienced by the fields, have been exploited in various con-texts, e.g. for atomic clock spectroscopy [17], magnetome-try [18], nonlinear and quantum optics [19]. In connectionwith the present work, it was predicted in [20–22] that thefree-space propagation in a resonant CPT medium would gen-erate pulses with matched statistics. Subsequently, the prop- agation of nonclassical quantum fluctuations in a coherentatomic medium under EIT or CPT conditions was investi-gated, both theoretically [23–26] and experimentally [27–32].The quantum properties of light fields interacting with a co-herent medium placed in an optical cavity was also analyzedin connection with quantum memory [33–35] and entangle-ment and spin-squeezing generation [36]. Here, the role ofthe cavity with respect to quantum state swapping is double,as it enhances the e ff ective optical depth of the medium - andthereby the swapping e ffi ciency - as well as provides a pas-sive mechanism for the exchange of the fluctuations in a well-defined frequency range within the cavity bandwidth. Since itis based on a quantum interference e ff ect intrinsically occur-ring between the fields in the atomic medium the device doesnot require additional quantum resources, such as entangle-ment, measurement or active feedback. The proposed mech-anism could also have applications in the microwave domain,e.g. with superconducting artificial atoms [42, 43]. |1 ñ |2 ñ |3 ñ A A |B ñ |D ñ |3 ñ A B A D FIG. 1. Atomic level structure considered: (a) Initial basis. (b)Dark / bright state basis. We consider an ensemble of N three-level Λ atoms withlong-lived ground / metastable states | i and | i and excitedstate | i . The atoms are positioned in an optical cavity, wherethey interact with two fields, A and A , resonant with the | i −→ | i and | i −→ | i transitions, respectively (fig. 1).The cavity is assumed single-ended, lossless and resonantwith both fields. Denoting by P i j ( i , j = −
3) the collectiveatomic operators, the interaction Hamiltonian in the rotating-wave approximation and in the rotating frame reads H int = − ~ ( g A P + g A P + h . c . ) , (1)where the g i ’s are the single-atom coupling strength. The evo-lution of the atom-field system is given by a set of Heisenberg-Langevin equations (see e.g. [34, 36])˙ P = − γ P + ig A ( P − P ) + ig A P + F , ˙ P = − γ P + ig A ( P − P ) + ig A P + F , ˙ P = − γ P − ig A P + ig A † P + F , ˙ P = γ P + ig A † P − ig A P + F , ˙ P = γ P + ig A † P − ig A P + F , ˙ A = − κ A + ig P + p κ A in , ˙ A = − κ A + ig P + p κ A in , where γ = ( γ + γ ) / γ theground-state coherence decay rate ( γ ≪ γ , ), κ and κ theintracavity field decay rates. The F i j ’s are zero-mean valuedLangevin noise operators, whose correlation functions can becalculated from the quantum regression theorem [37, 38]. A in and A in are the input field operators. By means of the stan-dard linearized input-output theory [37, 38], one can calculatefrom these equations the mean values of the observables insteady-state (denoted by h ξ i ), and derive the evolution equa-tions of the quantum fluctuations δξ = ξ − h ξ i . Going to theFourier space, it is then possible to relate the fluctuations ofthe fields exiting the cavity, A outj = p κ j A j − A inj ( j = , S X j ,θ ( ω ) at a given sideband fre-quency ω , where the quadrature fluctuations and noise spectraare standardly defined by δ X j ,θ = δ A j e − i θ + δ A † j e i θ and h δ X j ,θ ( ω ) δ X j ,θ ( ω ′ ) i = πδ ( ω + ω ′ ) S X j ,θ ( ω ) ( j = , . (2)This full quantum mechanical calculation can be performedwithout approximation for any Gaussian input field states.However, for the sake of the discussion and in order to de-rive analytical results, we will in the following focus on thesymmetric situation of fields with comparable intracavity Rabifrequencies, Ω i = g i h A i i ∼ Ω ( i = , ffi cient to saturatethe two-photon transition: Ω ≫ γγ . In this case, the atomsare pumped into a dark state |−i = ( | i − | i ) / √ h P i ∼ − N /
2. It is then convenient to turn tothe dark / bright state basis |−i , | + i , where | + i = ( | i + | i ) / √ bright and dark optical modes A ± = ( A ± A ) / √ h A − i =
0, and onefinds the situation analyzed in [34], in which the atoms, allin state |−i , are coupled to the empty dark mode A − on thetransition |−i −→ | i , and the bright mode A + on the transi-tion | + i −→ | i . The bright mode ”sees” no atoms, but in-duces electromagnetic transparency for the dark mode. Onecan then show that the equations for the fluctuations of thedark mode A − , of the ground-state coherence Q = |−ih + | andof the dark dipole P − = |−ih | are decoupled from those of thebright mode A + , and given by [34]( κ − i ω ) δ A − = ig δ P − + √ κδ A in − , (3)( γ − i ω ) δ P − = i Ω ′ δ Q + igN δ A − + F − , (4)( γ − i ω ) δ Q = i Ω ′ δ P − + F Q , (5)where Ω ′ = Ω √
2, we assumed g = g = g , κ = κ = κ , F − and F Q are zero-mean valued Langevin operators with correlation functions h F − ( ω ) F †− ( ω ′ ) i = γ N δ ( ω + ω ′ ) and h F Q ( ω ) F † Q ( ω ′ ) i = γ N δ ( ω + ω ′ ). Assuming a small ground-state decoherence rate ( γ ≪ γ, κ ), the fluctuations of the out-going dark mode are readily found to be δ A out − = κ + i ω − βκ − i ω + β δ A in − + F in , (6)where β ( ω ) = g N ( γ − i ω )( γ − i ω )( γ − i ω ) + Ω ′ , F in = √ κκ − i ω + β F − . (7)Since the transmission function of the bright mode fluctua-tions is that of an empty cavity, δ A out + = κ + i ωκ − i ω δ A in + , (8)one readily shows that the quadrature noise spectra of the out-going initial modes are given by S X out ,θ = | λ + + λ − | S X in ,θ + | λ + − λ − | S X in ,θ + − | λ − | , (9) S X out ,θ = | λ + + λ − | S X in ,θ + | λ + − λ − | S X in ,θ + − | λ − | , (10)where λ + = κ + i ωκ − i ω , λ − = κ + i ω − βκ − i ω + β . (11)The previous relations can be straightforwardly interpretedin terms of frequency-dependent swapping. For a largeenough cooperativity, C = g N / κγ , and not too high inten-sities, Ω ′ ≪ g √ N , the intracavity fields see a cavity with ane ff ective cavity halfwidth κ CPT ≃ γ + κ Ω ′ g N ! , (12)much narrower than the bare cavity halfwidth [11–16]. Onecan then distinguish three regimes depending on the sidebandfrequency considered: (i) a transparency regime for ω ≪ κ CPT , where the trans-mission of the atom-cavity system is that of a resonantempty cavity. The fluctuations of the outgoing fields arethen equal to those of the incoming fields, δ A out , = δ A in , ( β ∼ λ + ∼ λ − ∼ (ii) a swapping regime for κ CPT ≪ ω ≪ κ , in which thedark mode sidebands see an o ff -resonant cavity and aretherefore π -shifted, while those of the bright mode see aresonant cavity and remain unchanged. From eqs. (6,8),one thus easily obtains that δ A out ∼ δ A in , δ A out ∼ δ A in ( λ + ∼ λ − ∼ − (iii) a reflection regime for ω ≫ κ , in which the cavity trans-mission is that of an o ff -resonant cavity and the fluctu-ations of the outgoing fields are those of the reflectedfields, δ A out , = − δ A in , .While the conservation of the fluctuations either within thetransparency window and outside the cavity bandwidth arerather intuitive, the exchange of fluctuations in the swappingregion may be less so. In this frequency window the CPTmedium acts as frequency-dependent phase-plate for the fieldsidebands, and the dephasing is di ff erent for fluctuations ofthe outgoing dark and bright field modes. Indeed, while thebright mode sidebands see an empty cavity δ A outB ∼ δ A inB ,the dark mode sidebands see an o ff -resonant cavity and theintracavity field fluctuations vanish: δ A D ≃
0, and thereby δ A outD ∼ − δ A inD . This implies that the intracavity fluctua-tions of the initial modes are equal: δ A ∼ δ A . This ef-fect is reminiscent of the matched pulse propagation discussedin [20, 22] and of the oscillatory transfer of squeezing dis-cussed in [25, 26], which occur with fields propagating insingle-pass through a medium. However, the cavity interac-tion sets here a natural frequency boundary, namely the cav-ity bandwidth, for the atomic-induced interference e ff ects andprovides an automatic locking mechanism for the coherent ex-change of fluctuations. When the atomic absorption is negligi-ble the cavity containing the CPT medium thus plays the roleof a lossless frequency-dependent phase-plate for the quantumfluctuations of the fields. - Ω (cid:144) Κ N o r m a li ze dno i s e s p ec t r u m FIG. 2. (Color online) Noise spectra of the amplitude ( θ =
0) quadra-ture of the outgoing fields ( A : red, A : blue). The dashed lines in-dicate the amplitude-quadrature noise spectra of the incident fields,which are in a coherent and a 3dB amplitude-squeezed state, respec-tively ( S X in ,θ = = S X in ,θ = = . g √ N , γ, Ω , γ ) = (10 , . , . , × κ . In order to illustrate this behavior we choose the two inputfields to have equal intracavity Rabi frequencies and to be ina coherent and a squeezed state, respectively. Without loss ofgenerality, we assume field 2 to be in a broadband, minimaluncertainty, amplitude-squeezed state with a squeezing arbi-trarily fixed to -3 dB: S X in ,θ = ( ω ) = / S X in ,θ = π/ ( ω ) = κ = γ , C = Ω ′ ∼ γ/ √ ff ective cavity band-width is indeed much smaller than κ ( κ CPT /κ ≃ . ω ± /κ ≃ ± √ C ≃ ±
10 in this case). For the same
FIG. 3. (Color online) E ffi ciency η of the squeezing transfer fromfield 2 to 1, as a function of analysis frequency ω (in units of κ , logscale) and cooperativity C , in the same configuration as in Fig. 2.Parameters: ( γ, Ω , γ ) = (0 . , . , × κ . configuration of a coherent and a squeezed input fields, fig. 3shows the e ffi ciency of the squeezing transfer, η ≡ − S X out ,θ = − S X in ,θ = = | λ + − λ − | , (13)as a function of the sideband frequency ω and the cooperativ-ity parameter C . In the swapping region, the e ffi ciency rapidlyincreases with C and can be shown to scale as η ≃ κ κ + ω ω ω + κ CPT (14) ≃ + κ CPT /κ ! ( ω ∼ √ κκ CPT ) . (15)We checked the e ff ect of the ground-state decoherence us-ing the full numerical simulations. A non-negligible γ hastwo e ff ects: first, it induces a coupling between the dark andbright states, thus adding excess atomic noise at low side-band frequencies (as can be seen e.g. from eq. (7)). Thisexcess atomic noise can lead to the reduction or disappear-ance of the squeezing at low frequencies. Secondly, it reducesthe atomic coherence between the ground-states, thereby de-creasing the quantum interference e ff ects. We checked how-ever that the swapping e ffi ciency remained high as long asthe transparency window is much larger than the ground-statedecoherence rate. Generally, since the transparency windowis ultimately limited by the ground-state decoherence rate γ ,using long-coherence time ensembles in low-finesse cavitiesis thus preferable for obtaining a high e ffi ciency as well as alarge dynamical range for the swapping. W / k W /k FIG. 4. (Color online) E ffi ciency η of the squeezing transfer fromfield 2 to 1, as a function of the field Rabi frequencies Ω and Ω (in κ units). Parameters: ( g √ N , γ, γ , ω ) = (10 , . , − , . × κ . We also examined numerically the non-symmetric situationof fields with di ff erent Rabi frequencies Ω , Ω . For su ffi -ciently strong driving on both transitions the dark and brightatomic state become | D i = ( Ω | i − Ω | i ) / Ω ′ , | B i = ( Ω | i + Ω | i ) / Ω ′ (16) ( Ω ′ = q Ω + Ω ), with corresponding dark and bright fieldcombinations. Figure. 4 shows the e ffi ciency of the squeezingtransfer from field 2 to 1 as the respective Rabi frequencies ofthe two fields are varied. The other parameters are the same aspreviously and the e ffi ciency was obtained numerically with afull calculation. Similar transparency windows are observedas in the balanced Rabi frequency case, and e ffi cient transfer isobserved as long as the CPT window is larger than the groundstate decay rate and smaller than the bare cavity halfwidth.On also finds that the transfer of quantum fluctuations is moste ffi cient for fields with balanced Rabi frequencies ( Ω ∼ Ω ).Qualitatively, this can be explained by the fact that the CPT-induced atomic coherence is maximal in this case and inducesperfectly destructive interference for the dark mode sidebands.In an unbalanced situation the fluctuations are only partiallyexchanged between the initial fields. Similarly to the free-space interaction [26], one can show, by following e.g. themethod of [44], that the fluctuations of the initial field modescan be retrieved in suitable combination of modes, which arehowever di ff erent from the initial ones.To conclude, we propose to use a coherent atomic mediumin an optical cavity to achieve passive quantum state swappingbetween two optical fields. E ffi cient exchange of quantumfluctuations can be achieved for reasonable e ff ective opticaldepth in the bad cavity limit and when there is an apprecia-ble narrowing of the cavity linewidth due to CPT. In additionto quantum information processing in the optical domain, theproposed mechanism could also have valuable applications forcircuit QED in the microwave domain, e.g. with supercon-ducting artificial atoms [42, 43]. ACKNOWLEDGMENTS
The author is grateful to Michel Pinard for initiating thiswork and Ian D. Leroux for pointing out the phase-plate in-terpretation of the swapping, and acknowledges financial sup-port from the European STREP and ITN projects
PICC and
CCQED . [1] Braunstein S. L. and van Loock P., Rev. Mod. Phys. , 513(2005).[2] Cerf N. J., Leuchs G. and Polzik E. S., Quantum informationwith continuous variables of atoms and light , Imperial CollegePress (2007).[3] Ralph T. C. and Lam P. K., Nature Photon. , 671 (2009).[4] Braunstein S. L. and Kimble H. J., Phys. Rev. Lett. , 869(1998).[5] Furusawa A., Sørensen J. L., Fuchs C. A., Braunstein S. L.,Kimble H. J. and Polzik E. S., Science , 706 (1998).[6] Polkinghorne E. and Ralph T. C., Phys. Rev. Lett. , 2095(1999).[7] Takei N., Yonezawa H., Aoki T. and Furusawa A., Phys. Rev.Lett. , 220502 (2005).[8] Hammerer K., Sørensen A. S.and Polzik E. S., Rev. Mod. Phys. , 1041 (2010). [9] Furusawa A. and van Loock P., Quantum Teleportation and En-tanglement: A Hybrid Approach to Optical Quantum Informa-tion Processing , Wiley (2011).[10] Arimondo E. and Orriols G., Lett. Nuovo Cimento , 333(1976).[11] Lukin M. D., Fleischhauer M., Scully M. O. and VelichanskyV. L., Opt. Lett. , 295 (1998).[12] Hernandez G., Zhang J. and Zhu Y., Phys. Rev. A , 053814(2007).[13] Wu H., Gea-Banacloche J. and Xiao M., Phys. Rev. Lett. ,173602 (2008).[14] M¨ucke M., Figueroa E., Borchmann J., Hann C., Murr K., Rit-ter S., Villas-Boas C. J. and Rempe G., Nature , 755 (2010).[15] Kampschulte T., Alt W., Brakhane S., Ekstein M., Reimann R.,Widera A. Meschede D., Phys. Rev. Lett. , 155603 (2010). [16] Albert M., Dantan A. and Drewsen M., Nature Photon. , 633(2011).[17] Vanier J., Appl. Phys. B , 421 (2005).[18] Budker D., Gawlik W., Kimball D. F., Rochester S. M.,Yashchuk V. V. and Weis A., Rev. Mod. Phys. , 1153 (2002).[19] Fleischhauer M., Imamoglu A. and Marangos J. P., Rev. Mod.Phys. , 633 (2005).[20] Harris S. E., Phys. Rev. Lett. , 552 (1993).[21] Agarwal G. S., Phys. Rev. Lett. , 1351 (1993).[22] Fleischhauer M., Phys. Rev. Lett. , 989 (1994).[23] Fleischhauer M. and Lukin M. D., Phys. Rev. A , 022314(2002).[24] Dantan A., Bramati A. and Pinard M., Phys. Rev. A , 043801(2005).[25] Barberis-Blostein P. and Bienert M., Phys. Rev. Lett. ,033602 (2007).[26] Hu X. and Oh C. H., Phys. Rev. A , 053842 (2011).[27] Akamatsu D., Akiba K. and Kozuma M., Phys. Rev. Lett. ,203602 (2004).[28] Hsu M. T. L., H´etet G., Gl¨ock O., Longdell J. J., Buchler B.C., Bachor H.-C. and Lam P. K., Phys. Rev. Lett. , 183601(2006).[29] Appel J., Figueroa E., Korystov D., Lobino M. and Lvovsky A.I., Phys. Rev. Lett. , 093602 (2008).[30] Honda K., Akamatsu D., Arikawa M., Yokoi Y., Akiba K., Na-gatsuka S., Tanimura T., Furusawa A. and Kozuma M., Phys.Rev. Lett. , 093601 (2008).[31] Cviklinski J., Ortalo J., Laurat J., Bramati A., Pinard M. andGiacobino E., Phys. Rev. Lett. , 133601 (2008). [32] Arikawa M., Honda K., Akamatsu D., Nagatsuka S., Akiba K.,Furusawa A. and Kozuma M., Phys. Rev. A , 021605 (2010).[33] Lukin M. D. and Fleischhauer M., Phys. Rev. Lett. , 4232(2000).[34] Dantan A. and Pinard M., Phys. Rev. A , 043810 (2004).[35] Dantan A., Bramati A. and Pinard M., Europhys. Lett. , 881(2004).[36] Dantan A., Cviklinski J., Giacobino E. and Pinard M., Phys.Rev. Lett. , 023605 (2006).[37] Courty J.-M., Grangier P., Hilico L. and Reynaud S., Opt. Com-mun. , 251 (1991).[38] Hilico L., Fabre C., Reynaud S. and Giacobino E., Phys. Rev.A , 4397 (1992).[39] Lambrecht A., Coudreau T., Steinberg A. and Giacobino E.,Europhys. Lett. , 93 (1996).[40] Josse V., Dantan A., Vernac L., Bramati A., Pinard M. and Gi-acobino E., Phys. Rev. Lett. , 103601 (2003).[41] Gorshkov A. V., Andr´e A., Lukin M. D. and Sørensen A. S.,Phys. Rev. A , 033804 (2007).[42] Kelly W. R., Dutton Z., Schlafer J., Mooskerji B., Ohki T. A.,Kline J. S. and Pappas D. P., Phys. Rev. Lett. , 163601(2010).[43] Bianchetti R., Filipp S., Baur M., Fink J. M., Lang C., Ste ff enL., Boissonneault O., Blais A. and Wallra ff A., Phys. Rev. Lett. , 223601 (2010).[44] Josse V., Dantan A., Bramati A. and Giacobino E., J. Opt. B6