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= ¼ ivg 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 dx ðxÞ½cRðxÞ þ cLðxÞ rent experimental and theoretical interest both for nano- 2 2 photonics in the optical regime [13–19] and in the studies þ þcRðxÞþ þcLðxÞ ; (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 ( ) 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 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 ðxÞ þj0i 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 ðxÞ þ0i: 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 ½ þ; H^ o ¼0, 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 iHt 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 ðxÞcRðxÞ þj0i: (2) 1 iHt ffiffiffi c e jinieo ¼ p dx1dx2 spðx1 tÞ ðx2 tÞjx1;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 dx1dx2 eoðxc t;xdÞjx1;x2ieo; (4) [Fig. 1(b)] 2
x1þx2 y jouti¼j RRiþj LLiþj RLi; where xc 2 , xd x1 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 dx1dx2 RR x1;x2 1= 2 cR x1 ZZ Z y 1 cRðx2Þ0i denotes an outgoing two-photon state with two iHt ffiffiffi iHet þ e jiniee ¼ p dk1dk2 dxe jk1k2 i right-moving photons and PRR h RRj RRi 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 dx1dx2 eeðxc 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 t; xdÞ and eoðxc t; xdÞ are calculated in c ðxÞ¼i eið i =2Þx ð xÞ: (3) Ref. [30], and as a result we have c 1 2 ð1 þ Þ3 8 For ¼ 1, ðxÞ¼ spðxÞ, where spðxÞ is the wave func- P ð Þ¼ 1 þ ; RR;LL 4 ð1 þ Þ2 ð 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 ; 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 ðxÞ ð1= 2Þ½cRðxÞþcLð xÞ and co ðxÞ ð1= 2Þ RR RL LL y y the temporal duration of the incident photon pulse is ½cRðxÞ cLð xÞ . 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=@ ¼ i 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 dx ðxÞ½cyðxÞ þ c ðxÞ þ ; e 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^ =@ ¼ i dxco ðxÞ c ðxÞ: pulse passes through. This limit of 1 corresponds to a o @x o scenario where the transmission of the incident photon We note that H ¼ He þ Ho and ½He;Ho ¼0. Similarly, and the decay of the atom occur independently. In this we decompose the in state: ‘‘independent’’ scenario, PRR ¼ 0:5. Therefore, PRR > 0:5
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(a) (b) (c) Similarly, from Eqs. (4) and (5), we can derive 0.75 0.75 0.75 ZZ 1 1 1 0.5 0.5 0.5 P ¼ þ F dx dx f½ ðx t; x Þ RR;LL 4 8 8 1 2 eo c d 0.25 0.25 0.25 þ eoðxc t; xdÞ eeðxc t; xdÞ 0 0 0 0.01 0.1 1 10 100 0.01 0.1 1 10 100 0.01 0.1 1 10 100 þ 2 eeðxc t; xdÞ eoðxc t; xdÞg: (7) α α α
In Eq. (7), we see that PRR and PLL (with þ and signs, FIG. 2 (color online). (a) fPRR;PLL;PRLg in solid green, dashed red, and dotted blue lines for a two-level atom. respectively) are influenced in the same way by the photon cavity cavity cavity indistinguishability factor F. Thus, the most prominent (b) fPRR ;PLL ;PRL g in solid green, dashed red, and dotted classical classical classical feature of stimulated emission in terms of P > 0:5 can- blue lines. (c) fPRR ;PLL ;PRL g in solid green, dashed RR red, and dotted blue lines. not be attributed to photon indistinguishability. The last term in Eq. (7) contains overlap integrals be- tween the out state eoðxc t; xdÞ in the eo subspace— indicates an enhanced probability that both photons go to which is a product of the incident wave function c ðx tÞ the right and is therefore a direct indication of stimulated and a spontaneously emitted photon ðx tÞ—and the emission. sp out state ðx t; x Þ in the ee subspace. Since a single In Fig. 2(a), we indeed observe P > 0:5 in the inter- ee c d RR two-level atom cannot simultaneously absorb two photons, mediate region of . In fact, P maximizes to 2=3 at RR the two-photon wave function ðx t; x Þ in the ee ¼ 2—when the temporal decay rate of the incident ee c d subspace exhibits complex entangled structures in spatial, photon pulse is twice the atom’s spontaneous emission spectral, and temporal domains [15]. It is known that rate. Moreover, the enhancement of P can be observed RR the two-photon wave function in the ee subspace has a with other incident pulse shapes. For example, numeri- direct connection to the Bethe-ansatz wave functions that cally, we have found that an incident photon with half- have been commonly used to describe strongly correlated c ðxÞ¼ð2 2 2Þ1=4e ð Þ2x2=4 ð xÞ Gaussian wave function quantum systems [15]. Hence, here we refer to the effect attains a similar maximum of P 0:65 for 1:6. RR arising from the complex structure in eeðxc t; xdÞ as a To better understand the stimulated emission behavior in strong photon correlation effect. The form of Fig. 2(a), we consider the general expression of the three Eq. (7) therefore directly points to the role that the probabilities PRR, PLL, and PRL, for an arbitrary incident strong correlation induced by the atom plays in stimu- c wave function ðxÞ. From Eqs. (4) and (5), one can show lated emission in addition to the effect of photon that indistinguishability. The role of photon indistinguishability in stimulated Z emission is a topic of current interest. In Ref. [4], Sun PRL ¼ 0:5 0:25 dx1dx2 eoðxc t; xdÞ eoðxc t;xdÞ et al. studied this question by comparing two scenarios. In Z the first scenario, an N-photon state is sent into a para- 2 metric down-converter, and the amplification results in ¼ 0:5 0:25 dxc ðxÞ ðxÞ ; sp the creation of an N þ 1 photon state in the output. In the second scenario, the amplification process is instead emulated by combining the N-photon state with an extra where to obtain the first line, we note that the cross terms photon at a beam splitter. The authors then showed experi- between eoðxc t; xdÞ and eeðxc t; xdÞ happen to mentally that these two scenarios produce the same photon cancel, and in obtaining the second line we have used statistics. Thus, the crucial aspect of stimulated emission Eq. (4). We define a photon indistinguishability factor in their system is the indistinguishability between the photons. Z Motivated by their work, here we also consider an 2 alternative scenario where we replace the atom with a F dxc ðxÞ ðxÞ ; (6) sp side-coupled cavity. We emulate the stimulated emission process by injecting a single-photon pulse into the wave- guide while there is a photon in the cavity. Mathematically, which measures the indistinguishability between the inci- this system as well as the initial state can be described by y y dent single-photon pulse and the single-photon pulse from replacing ð þÞ with aða Þ and z with 2a a 1 in spontaneous emission of an excited atom. When ¼ 1, Eqs. (1) and (2). Here a (ay) is the annihilation (creation) y F maximizes at 1. Consequently, PRLð Þ is minimal. operator of a cavity photon, and ½a; a ¼1. Also, we note Moreover, since 0 F 1,wehave0:25 PRL 0:5, that the waveguide-atom and waveguide-cavity systems establishing an upper bound of 3=4 for PRR. have identical behaviors in the single-excitation manifold.
143602-3 week ending PRL 108, 143602 (2012) PHYSICAL REVIEW LETTERS 6 APRIL 2012 3 In the absence of the injected single photon in the wave- guide, the photon in the cavity decays spontaneously. Thus, 2 photon indistinguishability in this case refers to how close the photon pulse matches the spontaneously decayed pho- 1 ton pulse from the cavity, as characterized by the same F factor in Eq. (6). For the waveguide-cavity system, the 0 output probabilities for various outcomes are 0.01 0.1 1 3 10 100 α 1 2 1 Pcavity ¼ 1 þ ; RR;LL 4 ð1 þ Þ2 þ 1 FIG. 3. a ( c) vs photon decay parameter for an atom (cavity) in the solid (dashed) curve. cavity PRL ð Þ¼PRLð Þ; which we plot in Fig. 2(b). We compare the waveguide-cavity system to a classical the atom is minimal and equals a ¼ 0:75= . This corre- system consisting of a waveguide and an ancilla. We con- sponds to the condition for a maximum in the photon sider an incident single-photon pulse with a probability correlation term in Eq. (7), demonstrating that stimulated c 2 1 c emission in this system is associated with a shortening of spectrum j ðkÞj ¼ 2 2 , where ðkÞ is the 2 ðk Þ þð =2Þ the atom lifetime. In contrast, the lifetime of a cavity momentum-space representation of c ðxÞ in Eq. (3). The R c 1 dthayai is always longer than 1= , which it approaches ancilla emits a classical photon with probability 1=2 in 0 each direction and, moreover, reflects an incident photon from above as !1. 2 ð =2Þ2 It is interesting to note that the maximum probability with a probability spectrum jrkj ¼ 2 2 . The an- ðk Þ þð =2Þ PRR ¼ 2=3, achieved at ¼ 2, is equal to the maximum cilla therefore has the same intensity response function to a probability of cloning a quantum state in a universal quan- single photon as either an atom or a cavity. For this tum cloning machine [3,31]. However, the system here is classical 1 classical classical case we have PRR ð Þ¼2 1þ , PLL ð Þ¼ not a universal quantum cloning machine. If we inject a 1 1 classical 2 1þ , and PRL ð Þ¼1=2, as plotted in Fig. 2(c).We single photon in the even subspace, both of the outgoing see that the waveguide-cavity system has an enhanced photons will be in the even subspace. Moreover, the initial cavity probability PRR compared to the classical system single-photon wave function is not cloned in the outgoing classical PRR . Photon indistinguishability, which is unique to two-photon wave function. Despite this fact, we can con- the quantum system, certainly plays a role here. sider the stimulated emission process studied here by using cavity the language of photon cloning, if one is interested only in However, PRR ð Þ is always smaller than 0.5. Thus, the excess probability of having both outgoing photons prop- the outgoing direction of the photons and not the detailed agating to the right, i.e., the fact that PRRð Þ > 0:5, cannot photon wave function. Then, the photon cloning fidelity [3] be achieved in the waveguide-cavity system and, therefore, maximizes at ¼ 3, yielding a fidelity of ¼ PRRþ cannot be attributed to photon indistinguishability. It fol- 1=2PRL ¼ 0:8125. The photon-number amplification fac- lows that the key difference between the atom and the tor is maximal at ¼ 2, with a value of 2PRR ¼ 4=3. parametric down-converter in Ref. [4] is the photon corre- In addition to the coupling between the atom and wave- lation induced by the atom due to its inability to absorb guide, the atom can also couple into nonguided modes, more than one photon at a time. which are considered a loss mechanism. In this case we In the waveguide-atom system, the effect of stimulated define the factor as , where is the coupling þ ng ng emission is also manifested in the lifetime of the atomic rate into nonguided modes. Large factors of 0.9 and excitation. We define such a lifetime as Z higher have been experimentally measured with the same 1 1 geometry studied here [32–34]. In our analysis, a less-than- a dth z þ 1i: 2 0 unity factor can be accounted for by making the replace- With this definition, a spontaneously decaying atom yields ment ! i ng=2 in the loss-free results. For ¼ a ¼ 1= , as expected. In Fig. 3, we plot a as a function 0:9, the results are modified only slightly, with the maxi- of , which one may recall from Eq. (3) characterizes the mum of PRR now occurring at 2:04, resulting in temporal decay rate of the incident photon pulse. In the PRR 0:63. limit 1, the atom lifetime is identical to the sponta- In summary, we have provided a fully quantum- neous emission lifetime 1= . In this regime, the incident mechanical theoretical study of stimulated emission at its photon’s duration is extremely short, which has no effect most basic level, involving an incident single-photon pulse on the atom’s lifetime. 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