Phase-modulated electromagnetically induced transparency in a giant-atom system within waveguide QED
aa r X i v : . [ qu a n t - ph ] F e b Phase-modulated electromagnetically induced transparency in a giant-atom systemwithin waveguide QED
Wei Zhao, Yan Zhang, and Zhihai Wang ∗ Center for Quantum Sciences and School of Physics,Northeast Normal University, Changchun 130024, China (Dated: February 10, 2021)The nonlocal emitter-waveguide coupling, which gives birth to the so called giant atom, representsa new paradigm in the field of quantum optics and waveguide QED. In this paper, we investigatethe single-photon scattering in a one-dimensional waveguide on a two-level or three-level giant atom.Thanks to the natural interference induced by the back and forth photon transmitted/reflected atthe atom-waveguide coupling points, the photon transmission can be dynamically controlled by theperiodic phase modulation via adjusting the size of the giant atom. For the two-level giant-atomsetup, we demonstrate the energy shift which is dependent on the atomic size. For the driven three-level giant-atom setup, it is of great interest that, the interference effect between different atomictransition paths, can lead to a complete transmission window, analogous to the electromagneti-cally induced transparency and beyond the two-photon resonance mechanism, and the width of thetransmission valleys (reflection range) is tunable in terms of the atomic size. Our investigation willbe beneficial to the photon or phonon control in quantum network based on mesoscopical or evenmacroscopical quantum nodes involving the giant atom.
I. INTRODUCTION
In the field of quantum optics, the study of the inter-action between light and matter is one of the long-livedsubjects. Recently, the light-matter interaction in waveg-uide structures has attracted much attention, which leadsto lots of theoretical and experimental works in waveg-uide QED community [1–6], such as dressed or boundstates [7–12], phase transitions [13, 14], single-photondevices [15–17], exotic topological and chiral phenom-ena [18–23], where the wavelength of light (or microwavefield) is usually tens or hundreds of times larger thanthe size of the natural/artificial atoms constituting thematter [24–28]. Therefore, the light-matter interaction isusually modelled by the dipole approximation, where theatoms are regarded as point-like dipoles [29].However, in recent years, the artificial superconduct-ing transmon qubit coupled by the acoustic waves [30–32], of which the size is comparable to the wavelengthof the phonons, is named as “giant atom” and has beensuccessfully realized in experiments [33] . Alternatively,the giant-atom model can also be realized in the super-conducting transmission line setup, where the capaci-tive or inductive coupling allows more than one couplingpoints between the microwave field and the qubit [34, 35].Moreover, using the cold atomic system, a theoreticalscheme for the realization of giant atom has been pro-posed in dynamical state-dependent optical lattices [36].In the giant-atom community, a lot of new phenomenanot existing in the conventional small atomic systemhave been predicted, such as frequency dependent relax-ation [37], non-exponential decay [38–40], tunable boundstate [41, 42] as well as decoherence free subspace [43, 44] ∗ [email protected] (For a recent review, see Ref. [45]). The underlyingphysics behind these phenomena is the interference andretarded effect during the photon/phonon propagatingprocess between the different coupling points.On the other hand, the dynamical control of the single-photon transmission is a hot topic in constructing quan-tum networks [46–48], motivated by the photon basedquantum information processing. Photons provide a reli-able transmission of quantum information and the waveg-uide is often seen as photonic channels in quantum net-work, with atoms (or artificial atoms) acting as quan-tum nodes. Along this line, people have proposed lotsof schemes to realize single-photon device, in which thepropagating of the photon in channels is controlled on de-mand by adjusting the nature of the quantum nodes [15–17, 49–53]. Combined with the interference effect, it mo-tivates us to study how to modulate the single photonscattering by the giant atom.In this paper, we tackle this issue in a one-dimensionalwaveguide with a two-level or three-level giant atom.For the two-level atom setup, we find that the changein the size of the atom can control its energy shiftdue to the interference effect between the backwardand forward photon in the waveguide. In the driventhree-level giant-atom system, we demonstrate the con-trollable phase-modulated electromagnetically inducedtransparency (EIT) [54–58] physics, which also arisesfrom and can be controlled by the incorporation ofthe two interference effects induced by the interplay ofthe photon transmissions and atomic transitions, respec-tively. And then, the size of the giant atom, which servesas a controller, can be used to tune the width of the trans-mission valleys (reflection) in a periodical manner. Theunderlying physical principal is further revealed in theviewpoint of quantum open system based on the dressedstate representation.The rest of the paper is structured as follows: In Sec. II, FIG. 1. Schematic configuration for a linear waveguide cou-pled to a giant atom at the points x = 0 and x = x . we present the model and discuss the single photon scat-tering in a two-level giant-atom system. In Sec. III, wedemonstrate the EIT scattering behavior for a driventhree-level giant-atom with Λ-type transition. At last,we end up with a brief conclusion in Sec. IV. II. TWO-LEVEL GIANT ATOM
As schematically shown in Fig. 1, the system we con-sider is composed of a linear waveguide and a giant atom,which is actually a two-level system. The giant atom isconnected to the waveguide via two points with x = 0 and x = x , respectively. The Hamiltonian H of the systemcan be divided into three parts, i.e., H = H s + H ω + V .The first part H s is the free Hamiltonian of the giantatom (Hereafter, we set ~ = 1). H s = ω e | e i h e | , (1)where ω e is the transition frequency between the groundstate | g i and the excited state | e i .The second part H ω of the Hamiltonian H representsthe free Hamiltonian of the waveguide, and is expressedas H ω = Z dx {− iv g C † R ( x ) ddx C R ( x )+ iv g C † L ( x ) ddx C L ( x ) } , (2)where v g is the group velocity of photons traveling inthe waveguide. Here, we assume that the group veloc-ity possesses the same unit as the frequency by consid-ering the length of the waveguide to be dimensionless. C † R ( x )[ C † L ( x )] is the bosonic creation operator for theright-going (left-going) photon at position x .For the third part V of the Hamiltonian H , we de-scribe the interaction between the waveguide and the gi-ant atom. Within the rotating wave approximation, the Hamiltonian V can be expressed as V = f Z dxδ ( x ) (cid:2) σ + C R ( x ) + σ + C L ( x ) + H . c . (cid:3) + f Z dxδ ( x − x ) (cid:2) σ + C R ( x ) + σ + C L ( x ) + H . c . (cid:3) , (3)where f is the coupling strength between the waveguideand the two-level giant atom. σ + = ( σ − ) † = | e ih g | isthe raising operator of the atom. The Dirac- δ functionin the Hamiltonian V indicates that the giant atom hasa length of x and connects to the waveguide via its headand tail, that is, x = 0 and x = x .It is noted that, the total excitation of the atom andthe photon in the waveguide is conserved. In the follow-ing section, we will restrict ourselves in the single exci-tation subspace, to investigate how to control the singlephoton scattering state via adjusting the frequency of thephoton and the size of the two-level giant atom.In the single-excitation subspace, the eigenstate of thesystem can be written as | E i = Z dx h φ R ( x ) C † R ( x ) + φ L ( x ) C † L ( x ) i | G i + u e σ + | G i , (4)where | G i represents that the waveguide is in the vac-uum states while the giant atom is in the ground state | g i . φ R ( x ) and φ L ( x ) are single-photon wave functionsof the right-going and left-going modes in the waveguide,respectively. u e is the excitation amplitude of the gi-ant atom. Solving the stationary Sch¨odinger equation H | E i = E | E i , the amplitudes equation can be obtainedas − iv g ddx φ R ( x ) + f u e M = Eφ R ( x ) , (5a) iv g ddx φ L ( x ) + f u e M = Eφ L ( x ) , (5b) ω e u e + f N = Eu e . (5c)where M = δ ( x ) + δ ( x − x ) and N = φ R (0) + φ R ( x ) + φ L (0) + φ L ( x ).Next, we consider the scattering behavior when a singlephoton with wave vector k is incident from the left side ofthe waveguide. In this case, the wave function of φ R ( x )and φ L ( x ) can be expressed as φ R ( x ) = e ikx { θ ( − x ) + A [ θ ( x ) − θ ( x − x )]+ t θ ( x − x ) } , (6) φ L ( x ) = e − ikx { r θ ( − x ) + B [ θ ( x ) − θ ( x − x )] } , (7)with θ ( x ) = x > x = 00 x < . (8) FIG. 2. The transmission rate T as functions of x and ∆.The parameters are set as ω e /v g = 20, f/v g = 0 . We explain the expression of the above wave functionsphysically as follows. When the right-going photon inci-dent from the region x < x = 0 between the giant atom and the waveguide,it can be transmitted or reflected, with the amplitudes of A and r , respectively. The photon transmitted at thefirst connection point at x = 0 will travel freely in thewaveguide until it reaches the second connection point at x = x , it will be then reflected or transmitted secondly,with the amplitudes B and t , respectively.Now, we substitute Eqs. (6-8) into the amplitudeEqs. (5), it yields the dispersion relation E = v g k . Fur-thermore, the transmission rate T = | t | can be ob-tained as T = (cid:0) ∆ v g − f sin kx (cid:1) (∆ v g − f sin kx ) + 4 f (1 + cos kx ) , (9)where ∆ = E − ω e is the detuning between the atom andthe propagating photon in the waveguide.In the small atom scenario ( x = 0) which is studiedin Ref. [15], it is obvious that the incident photon will becompletely reflected ( T = 0) when it is resonant with theatom, that is, ∆ = 0. However, for the giant atom in oursetup, the incident photon will propagate back and forthin the spatial regime covered by the giant atom, leadingto an interference effect. As a result, as shown in Fig. 2,the transmission rate can be controlled by adjusting thephoton-atom detuning ∆ and the size of the giant atom x . Interestingly, in the giant-atom situation ( x = 0),the detuning for complete reflection is determined by thetranscendental equation∆ r = 2 f sin k r x v g , (10)where ∆ r is dependent on k r via ∆ r = v g k r − ω e . Sincethe atomic frequency is implied by the location of the (b)(c) FIG. 3. (a) The detuning ∆ r as a function of x for f/v g =0 .
1. The solid line is given by the solution of the transcenden-tal equation in Eq. (10) and the empty circles are the resultsfrom the numerical fitting. (b) and (c) The fitting parameters S and K in Eq. (11) as a function of the atom-waveguide cou-pling strength f . In all of the panels, the atomic transitionfrequency is set to be ω e /v g = 20. complete reflection of the incident photon [15], we canobserve an atomic frequency shift which originates fromthe interference effect as discussed above, compared withthe small atom system. It implies that, only when thefrequency of the incident photon satisfies | ∆ r | ≤ f /v g ,that is, ω e +2 f /v g ≥ v g k r ≥ ω e − f /v g , it is possible tobe completely reflected. Within this regime, we plot thedependence of ∆ r on the size of giant atom x in Fig. 3(a)(solid line), which shows a sinusoidal shape. The exactsinusoidal shape by the numerical fitting (empty circles)is also shown in the figure and the fitting function isobtained by ∆ r ≈ S sin( Kx ) , (11)where the fitting parameters S ≈ f /v g and K are plot-ted as a function of atom-waveguide coupling strength f in Fig. 3(b) and (c), respectively. For the weak coupling,it shows that K is nearly equal to ω e /v g ( ω e /v g = 20in our consideration) and gradually diverges from it ina quadratic manner. Therefore, the energy shift of thegiant atom can be larger than its natural line width (notconsidered in this paper) and observed experimentally byincreasing the atom-waveguide coupling strength. III. THREE-LEVEL GIANT ATOM
In this section, let us consider the single-photon scat-tering on a driven Λ-type three-level giant atom. Asschematically shown in Fig. 4, the atom is characterizedby the ground state | g i , excited state | e i and metastablestate | f i . Here, there are two types of interference chan-nels. One is similar to that in two-level giant atom sys-tem, that is, the interplay between the multiple back-ward and forward photons of the transmission and re-flection in the waveguide; the other is peculiar for sucha three-level EIT atomic system, that is, the interferencebetween two transition paths | g i → | e i and | g i → | e i →| f i → | e i . The incorporation between two interferencechannels may provide a possibility of the extra and in-teresting property of the giant-atom EIT effect.The Hamiltonian H of the current system can be di-vided into three parts, i.e., H = H w + H a + V . Here, H w is the free Hamiltonian of the waveguide which is givenin Eq. (2). H a is the Hamiltonian of the giant atom H a = ω e | e i h e | + ω f | f i h f | + η ( | e i h f | e − iω d t + | f i h e | e iω d t ) , (12)where ω e and ω f ( ω f < ω e ) are the frequencies of thestate | e i and | f i , respectively. As a reference, we haveset ω g = 0. η and ω d are, respectively, the strength andfrequency of the classical field, which drives | f i ↔ | e i transition. In the rotating frame, the time independentHamiltonian becomes˜ H a = ω e | e i h e | + δ | f i h f | + η ( | e i h f | + | f i h e | ) , (13)where δ = ω f + ω d .The third part V of the Hamiltonian H describes theinteraction between the waveguide and the giant atom.Within the rotating wave approximation, it can be ex-pressed as V = f Z dxδ ( x ) nh C † R ( x ) + C † L ( x ) i | g i h e | + H . c . o + f Z dxδ ( x − x ) nh C † R ( x ) + C † L ( x ) i | g i h e | + H . c . o , (14)where f is the coupling strength between the waveguideand giant atom.In the single-excitation subspace, the eigenstate of thesystem can be expressed as | E i = Z dx h φ R ( x ) C † R + φ L ( x ) C † L i |∅ , g i + λ e |∅ , e i + λ f |∅ , f i , (15) FIG. 4. Schematic configuration for a linear waveguide cou-pled to a Λ-type three-level giant atom at the points x = 0and x = x . where |∅ , m i ( m = e, g, f ) represents that the waveguideis in the vacuum state while the giant atom is in thestate | m i . φ R ( x ) and φ L ( x ) are single-photon wavefunctions of the right-going and left-going modes in thewaveguide, respectively. λ e and λ f are the excitationamplitudes of the giant atom in the excited state | e i andmetastable state | f i , respectively. Similar to the discus-sion in two-level atom setup, for a right-going incidentphoton with wave vector k , the photon amplitude can beexpressed as φ R ( x ) = e ikx { θ ( − x ) + A [ θ ( x ) − θ ( x − x )]+ t θ ( x − x ) } , (16) φ L ( x ) = e − ikx { r θ ( − x ) + B [ θ ( x ) − θ ( x − x )] } , (17)where t and r are the transmission and reflection am-plitudes while A ( B ) are the amplitudes for findinga right (left)-going photon inside the regime of the gi-ant atom. Then, the Sch¨odinger equation ˜ H | E i = E | E i (where ˜ H = H w + ˜ H a + V ) yields t = ( E − δ ) (cid:2) i ( E − ω e ) v g − if sin ( kx ) (cid:3) − iv g η ( E − δ ) [ i ( E − ω e ) v g − f (1 + e ikx )] − iv g η , (18) r = 2 f ( E − δ ) (1 + cos ( kx )) e ikx ( E − δ ) [ i ( E − ω e ) v g − f (1 + e ikx )] − iv g η . (19)Then the transmission rate T = | t | and reflection rate R = | r | satisfy T + R = 1 due to the neglect of thenatural relaxations in the atom.In Fig. 5, we plot the transmission rate T as functionsof the detuning ∆ = E − ω e between the incident pho-ton and atomic | g i ↔ | e i transition and the size of thegiant atom x . Here, we illustrate the result for reso-nantly driving the atom by the field η in Fig. 5(a) and(d), and non-resonantly in Fig. 5(b),(c) and (e). All ofthe results are characterized by two narrow transmission -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 100.51-2 -1.5 -1 -0.5 0 0.5 1 1.5 200.51 (d) (e) FIG. 5. The transmission rate T as functions of the detuning ∆ = E − ω e and the size of the giant atom x for resonantdriving. The parameters are set as ω e /v g = 20, ω f /v g = 15, f/v g = 0 . η/v g = 0 .
5. The other parameters are ω d /v g = 5, ω d /v g = 6, ω d /v g = 4 for (a), (b) and (c), respectively. (d) The rate T for the lines labelled by “A” (solid) and “B” (dashed)in(a). (e) The rate T for the line labelled by “C” (solid) and “D” (dashed) in (b) and (c), respectively. valleys ( T ≃
0) and a relatively wide transmission win-dows ( T ≃
1) between them around the two-photon res-onance.First, we focus on the transmission window ( T ≃ , R ≃ R =0when E − δ = E − ω f − ω d = 0, which implies the two-photon resonance condition ω e − E = ω e − ω f − ω d . Itis same to that in the small atom setup and the com-plete transmission is popularly regarded as induced bythe destructive interference between the | g i → | e i and | g i → | e i → | f i → | e i transition paths. This is similarto the usual EIT phenomenon [54–57]. However, we didnot consider the natural relaxations of the states | e i and | f i , therefore, the window (valley) here is much wider(narrower) than that widely studied in EIT. Moreover,we can also observe that R = 0 when 1 + cos kx = 0,that is kx = (2 m + 1) π where m is an integer, and thedependence of the position is peculiar for the giant atomsetup. Note that kx is the accumulated phase as thetravelling photon moves from one coupling point to theother. Therefore, such a EIT-like transmission is also re- lated to the position-dependent phase introduced by theinterference effect from the back and forth photons insidethe regime covered by the giant atom. Next, we discussthe two valleys, which represent the complete reflection( T ≃ , R ≃ E ± = ω e + f sin( kx ) /v g − ∆ ± q (∆ + 2 f sin ( kx ) /v g ) + 4 η , (20)where ∆ = ω e − ω f − ω d is the detuning between the driv-ing field and the atomic | f i ↔ | e i transition. This factcan be explained intuitively in the dressed state presen-tation. The eigenfrequencies of the driving Hamiltonian˜ H a in Eq. (13) are ω ± = ω e − ∆ ± p ∆ + 4 η . (21) TABLE I. The values of x , ∆ and corresponding values of | G ± ,k | for the two valleys in Fig. 5 (d) and (e).Line Label curve ∆ /v g x f | G − ,ω − /v g | /v g f | G + ,ω + /v g | /v g A solid in Fig. 5(d) 0 0 . . . .
864 1 . . − .
984 1 . . .
984 0 . . and the corresponding states are | ψ + i = cos θ | e i + sin θ | f i , (22) | ψ − i = − sin θ | e i + cos θ | f i . (23)For the resonantly driving (∆ = 0), we will have θ = π/
2. For the case of non-resonantly driving, we will have θ = atan(2 η/ ∆ ) for ∆ > θ = π + atan(2 η/ ∆ )for ∆ < ω ± . This fact is verified by the re-sults shown in Fig. 5. Here, we use the phrase “nearly” toimply that E ± is not exactly equal to ω ± , but is slightlymodulated by x . As a result, the photon transmissionshows a periodic phase modulation in terms of x , whichis clearly demonstrated in Figs. 5 (a), (b) and (c) for bothof resonantly and non-resonantly driving situations.It is also shown in Fig. 5 that the two valleys discussedabove possess different widths. For example, see the hor-izontal lines labelled by “A”, “B”, “C” and “D” in Figs. 5(a), (b) and (c). This can be explained by the differenteffective decay rates of the eigenstates with frequency ω ± in the viewpoint of quantum open system by regardingthe waveguide as the effective environment. To this end,we rewrite the interaction Hamiltonian V in the momen-tum space by performing the Fourier transformation (interms of | ψ ± i ) as V = X k { [ C † L ( k )+ C † R ( k )] | g i [ G + ,k h ψ + | + G − ,k h ψ − | ]+H . c . } (24)where G + ,k = f cos θ e ikx ) , G − ,k = − f sin θ e ikx ) , (25)characterize the coupling strength between the k th modein the waveguide and the giant atom. We note that, | G + ,k | and | G − ,k | actually reflect the width of the twovalleys when the value of k is taken to satisfy v g k = ω ± .In Table I, we list the values of | G + ,k | and | G − ,k | for thehorizontal lines in Fig. 5(a) (b) and (c), along which thetransmission rates are plotted as functions of the detun-ing ∆ in Fig. 5 (d) and (e). It shows a good agreement forthe valley width between the table and the curves. Forthe resonant driving, which valley is wider depends on the size of the giant atom x , as shown in Fig. 5 (a) and(d). However, for the non-resonant driving, as shown inFig 5 (c), (c) and (e), the left valley is nearly always nar-rower than the right one when ∆ > <
0. In this sense, the modification to the photontransmission by the atomic size is more sensitive for theresonant driving setup.
IV. CONCLUSION
In this paper, we have studied the single-photon trans-mission in a one-dimensional linear waveguide systemcoupled with two-level or three-level giant atom. Gen-erally, in the giant-atom regime, the backward and for-ward photons propagating in the waveguide can lead toan interference effect. Thus, for the case of two-level gi-ant atom, we have shown that the complete reflectionoccurs, but not for the case that the incident photon isexactly resonant to the atom. In other words, the nat-ural interference leads to an effective small but nonzerofrequency shift, and the shift can be approximately re-garded as a sinusoidal function of the atomic size. For thedriven three-level giant atom, with the other interferencechannel induced by the two different atomic transitionpaths, we obtain a EIT-like line shape; more, with the in-corporation between these two interference channels, thetransparency window can be controlled by phase modu-lation in terms of the size of the atom. Also, the locationand width of the transmission valleys are tunable by ad-justing the atomic size.Beyond the specific model here, our work demonstrateshow to use the interplay between (among) the differentinterference channels to modify the photon transmissionin the waveguide. In the giant atom scenario, one of theinterference channels is provided by the photon propaga-tion, while the other arises from the atomic transitions.Motivating by the interplay mechanism, we hope that thegiant atom can be useful in designing photon or phononbased quantum device, which goes beyond the conven-tional small atom setup.
ACKNOWLEDGMENTS
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