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Stimulated Emission from a Single Excited Atom in a Waveguide

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Stimulated Emission from a Single Excited Atom in a Waveguide week ending PRL 108, 143602 (2012) PHYSICAL REVIEW LETTERS 6 APRIL 2012 Stimulated Emission from a Single Excited Atom in a Waveguide Eden Rephaeli1,* and Shanhui Fan2,† 1Department of Applied Physics, Stanford University, Stanford, California 94305, USA 2Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA (Received 6 September 2011; published 3 April 2012) We study stimulated emission from an excited two-level atom coupled to a waveguide containing an incident single-photon pulse. We show that the strong photon correlation, as induced by the atom, plays a very important role in stimulated emission. Additionally, the temporal duration of the incident photon pulse is shown to have a marked effect on stimulated emission and atomic lifetime. DOI: 10.1103/PhysRevLett.108.143602 PACS numbers: 42.50.Ct, 42.50.Gy, 78.45.+h Introduction.—Stimulated emission, first formulated by In recent years, there has been much advancement in Einstein in 1917 [1], is the fundamental physical mecha- the capacity to deterministically generate single photons nism underlying the operation of lasers [2] and optical [25,26] and to control the shape of the single-photon pulse amplifiers [3,4], both of which are of paramount impor- [27,28]. In both waveguide and free space, a single photon, tance in modern technology. In recent years, stimulated by necessity, must exist as a pulse [29]. Therefore, our emission has been studied in a variety of novel systems, study of stimulated emission at the single-photon level including a surface plasmon nanosystem [5], a single mole- reveals important dynamic characteristics of stimulated cule transistor [6], and superconducting transmission lines emission. For instance, we will show that there exists an [7]. Stimulated emission also plays a crucial role in the optimal spectral bandwidth of the single-photon pulse that quantum cloning of photons [3,8–10]. results in the largest enhancement of forward scattered In textbooks [11], stimulated emission is usually de- light. This is fundamentally different from the conven- scribed by having a photon interact with an atom in the tional study of stimulated emission involving a single- excited state. As a result of such interaction, the atom emits mode continuous radiation field. Finally, our results show a second photon that is ‘‘identical’’ to the incident photon. that, in the waveguide-atom system, emission enhance- As a concrete experimental example, in Refs. [4,12], pho- ment in the stimulated process cannot be entirely attributed tons of a given polarization were sent into a parametric to photon indistinguishability but, instead, has a substantial amplifier, and stimulated emission manifested in an en- connection to the strong photon correlation induced by the hanced probability of the outgoing photons having the two-level atom. same polarization. In Refs. [4,12], such enhancement was Model system and the incident single-photon pulse.—We attributed entirely to constructive interference due to pho- consider the atom-waveguide system shown in Fig. 1.Fora ton indistinguishability. waveguide with a linear dispersion relation, the real-space In this Letter, we consider a scenario of stimulated Hamiltonian of the system in the rotating-wave approxi- emission that is arguably closest to the textbook descrip- mation is [13] tion [11]. We consider an atom coupled to a waveguide and Z Z ^ @ y @ y @ study the interaction of the atom with a single incident H= ¼ivg dxc ðxÞ cRðxÞþivg dxc ðxÞ cLðxÞ R @x L @x photon, when the atom is initially in the excited state, as Z shown in Fig. 1(a). The atom-waveguide system is of cur- Vffiffiffi y y þ z þ p dxðxÞ½cRðxÞÀ þ cLðxÞÀ rent experimental and theoretical interest both for nano- 2 2 photonics in the optical regime [13–19] and in the studies þ þcRðxÞþþcLðxÞ; (1) of superconducting transmission lines in the microwave regime [20,21]. In an atom-waveguide system, the ability where x is the spatial coordinate along the waveguide’s y y to amplify few-photon quantum states is important and has symmetry axis. cRðxÞ [cLðxÞ] creates a right- [left-] moving been recently demonstrated experimentally [7]. Thus, the photon, cRðxÞ [cLðxÞ] annihilates a right- [left-] moving scenario that we consider is of experimental interest. photon, and þ (À) is a raising (lowering) operator of the We observe that the Hamiltonian describing the two-level atom. The atom-waveguide coupling strength is waveguide-atom system also describes, in 3D, the interac- V, corresponding to an atom spontaneous emission rate 2 tion of a light beam with an atom, when the beam is À V =vg. The transition frequency of the atom is . vg designed to mode-match the atom’s radiation pattern. is the waveguide group velocity, which we set as vg ¼ 1 This observation was emphasized and exploited in a num- from here on out. ber of recent papers [6,22–24]. Thus, our result should be At t ¼ 0, we assume that the atom is in the excited state. relevant to a wide class of recent 3D experiments as well. At the same time, a right-going single-photon pulse starts 0031-9007=12=108(14)=143602(5) 143602-1 Ó 2012 American Physical Society week ending PRL 108, 143602 (2012) PHYSICAL REVIEW LETTERS 6 APRIL 2012 jini¼jinieo þjiniee ZZ 1ffiffiffi c y ¼ p dx1dx2 ðxÞco ðxÞþj0i 2 FIG. 1 (color online). Schematic of the atom-waveguide sys- ZZ 1ffiffiffi c y tem studied here. (a) The initial state at t ¼ 0, consisting of an þ p dx1dx2 ðxÞce ðxÞþ0i: incident single-photon pulse and an atom in the excited state. 2 (b) At t !1the atom is in the ground state, and the photon field We note that, since ½þ; H^ o¼0, the state þj0i is con- is in a superposition of two-photon states j RRi, j LLi, and j i (green, red, and blue, respectively). tained in the even subspace. RL We can now calculate the out state by evaluating ÀiHt to interact with the atom. The in state of the system is limt!1e jini. From this point on, with the variable t therefore we always assume that we are in the t !1 limit. We Z consider jinieo and jiniee separately: c y ZZ jini¼ dx ðxÞcRðxÞþj0i: (2) 1 ÀiHt ffiffiffi c e jinieo ¼ p dx1dx2spðx1 ÀtÞ ðx2 ÀtÞjx1;x2ieo 2 At t !1the atom has decayed to the ground state, and the ZZ 1ffiffiffi out state contains only two-photon states, as given by p dx1dx2eoðxc Àt;xdÞjx1;x2ieo; (4) [Fig. 1(b)] 2 x1þx2 y jouti¼jRRiþjLLiþjRLi; where xc 2 , xd x1 À x2, and jx1;x2ieo ¼ ce ðx1Þ y R pffiffiffi co ðx Þj0i. Similarly, j i¼ ð Þð Þ yð Þ 2 where e.g., RR dx1dx2RR x1;x2 1= 2 cR x1 ZZ Z y 1 cRðx2Þ0i denotes an outgoing two-photon state with two ÀiHt ffiffiffi ÀiHet þ e jiniee ¼ p dk1dk2 dxe jk1k2 i right-moving photons and PRR hRRjRRi is its respec- 2 tive probability. Here, we use to denote all outgoing þ c y y hk2 k1j ðxÞce ðxÞa j0i states and use subscripts to distinguish these various states. ZZ Since the incident photon is right-moving, to study stimu- ¼ dx1dx2eeðxc À t; xdÞjx1;x2iee; (5) lated emission we will be particularly interested in com- paring behaviors of PRR, where both outgoing photons where jk kþi is the two-excitation interacting eigenstate are right-moving, to other outcomes as described by P 1 2 RL of H^ in the two-excitation manifold, as determined in and P . e pffiffiffi LL j i ¼ð Þ yð Þ yð Þj i For computation in this Letter, we choose an input wave Ref. [15], and x1;x2 ee 1= 2 ce x1 ce x2 0 . function For the specific initial state as defined in Eq. (2), pffiffiffiffiffiffiffi eeðxc À t; xdÞ and eoðxc À t; xdÞ are calculated in c ðxÞ¼i Àeið ÀiÀ=2ÞxðxÞ: (3) Ref. [30], and as a result we have c 1 2 ð1 þ Þ3 À 8 For ¼ 1, ðxÞ¼spðxÞ, where spðxÞ is the wave func- P ð Þ¼ 1 þ Æ ; RR;LL 4 ð1 þ Þ2 ð À 1Þð1 þ Þ2 tion of a photon spontaneously emitted by the excited atom. Variation of allows us to probe various time and 1 2 P ð Þ¼ 1 À ; frequency scales, as well as the connection between spon- RL 2 ð1 þ Þ2 taneous and stimulated emission in radiation dynamics. The out state.—To calculate the out state for an arbitrary which we plot in Fig. 2(a). initial statep asffiffiffi described by Eq. (2), it is useful to definepffiffiffi Analysis of stimulated emission.—When 1, y y y y P ð Þ!0, and P ð Þ, P ð Þ!0:5. In this limit, ce ðxÞð1= 2Þ½cRðxÞþcLðxÞ and co ðxÞð1= 2Þ RR RL LL y y the temporal duration of the incident photon pulse is ½cRðxÞcLðxÞ. With these definitions, the Hilbert space is separated into even and odd subspaces, each described much longer than the spontaneous emission lifetime. by a Hamiltonian: Thus, the initial excitation of the atom decays spontane- ously, and the pulse is completely reflected by an atom Z @ y largely in the ground state [13]. When 1, PRRð Þ, H^ e=@ ¼i dxce ðxÞ ceðxÞ @x P ð Þ!0:5, and P ð Þ!0. In this case, the photon Z RL LL pulse is extremely short; there is therefore no atom-photon þ V dxðxÞ½cyðxÞ þ c ðxÞ þ ; e À þ e 2 z interaction. As a result, the incident photon pulse is fully Z transmitted, and the atom decays spontaneously after the y @ H^ =@ ¼i dxco ðxÞ c ðxÞ: pulse passes through.
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