aa r X i v : . [ qu a n t - ph ] J a n Elliptical rotation of cavity amplitude in ultrastrong waveguide QED
Kazuki Koshino College of Liberal Arts and Sciences, Tokyo Medical and Dental University, Ichikawa, Chiba 272-0827, Japan (Dated: January 12, 2021)We investigate optical response of a linear waveguide quantum electrodynamics (QED) system,namely, an optical cavity coupled to a waveguide. Our analysis is based on exact diagonalizationof the overall Hamiltonian and is therefore rigorous even in the ultrastrong coupling regime ofwaveguide QED. Owing to the counter-rotating terms in the cavity-waveguide coupling, the motionof cavity amplitude in the phase space is elliptical in general. Such elliptical motion becomesremarkable in the ultrastrong coupling regime due to the large Lamb shift comparable to the barecavity frequency. We also reveal that such elliptical motion does not propagate into the outputfield and present an analytic form of the reflection coefficient that is asymmetric with respect to theresonance frequency.
I. INTRODUCTION
Cavity quantum electrodynamics (QED) deals with the interaction between a single atom and a discretized photonmode confined in a resonator, which is the simplest embodiment of quantum light-matter interaction. The cavity QEDsystems have been realized in various physical platforms: just to cite a few, single atoms coupled to an optical cavity,a semiconductor quantum dot in a photonic-crystal cavity, and a superconducting qubit coupled to a transmission-lineresonator. Interestingly, regardless of its physical platform, a cavity QED system is characterized by several universalparameters, such as ω a and ω c (atom and cavity frequencies), g (atom-photon coupling), κ (cavity decay rate), and γ (atomic decay rate into environments). In the history of cavity QED, extensive efforts have been made to reach thestrong-coupling regime ( g > κ, γ ), where the vacuum Rabi oscillation and splitting become observable [1–4]. In usualstrong-coupling systems, the coupling is still by far smaller than the resonance frequencies of the atom and cavity.Recently, attainments of the ultrastrong-coupling ( g & ω a,c /
10) and deep-strong-coupling ( g & ω a,c ) regimes havebeen reported [5–11]. In such ultrastrong-coupling systems, the counter-rotating terms in the Hamiltonian, which donot conserve the total number of excitations and are usually negligible in the weakly coupled systems, result in severalintriguing physical phenomena, such as the Bloch-Siegert shift [12, 13], virtual photons in the ground state [14–18],and the number non-conserving optical processes such as multiphoton vacuum Rabi oscillation [19–21].Waveguide QED deals with the interaction between a single atom and a one-dimensional continuum of photonmodes, typically provided by a waveguide attached to the atom. The parameters to characterize waveguide QEDsystems are ω a , γ e (atomic decay rate into waveguide) and γ i (atomic decay rate into environments). The strong-coupling regime in waveguide QED is defined by γ e > γ i , namely, the condition that radiation from the atomis dominantly forwarded to the waveguide [22–27]. This is reflected in spectroscopy as a strong suppression oftransmission near the atomic resonance. Following the definitions in cavity QED, the ultra- and deep-strong waveguideQED should be defined as γ e & ω a /
10 and γ e & ω a , respectively. The ultrastrong and deep-strong regimes of waveguideQED have already been reached using a superconducting qubit [28, 29]. Theoretically, up to the usual strong-coupling regime, perturbative treatment of dissipation based on the rotating-wave and Born-Markov approximationsprovides convenient and powerful theoretical tools, such as the Lindblad master equation and the input-outputformalism [30, 31]. However, this is not the case in highly dissipative regimes, and rigorous numerical methods areactively developed [32–35].In this study, we investigate a linear waveguide QED setup, namely, a harmonic oscillator coupled to a waveguide,and investigate its optical response to a classical drive field applied through this waveguide. A merit of this system is r waveguide r = cavity drive field FIG. 1: Schematic of a cavity-waveguide system. A cavity is coupled to a semi-infinite waveguide, through which a monochro-matic drive field is applied. The r < r >
0) region in the waveguide corresponds to the input (output) port. that the overall Hamiltonian is diagonalizable by the Fano’s method [36–38] and rigorous optical response is accessibleeven for highly dissipative situations. We report an elliptic motion of the oscillator in the phase space, which occurs,in principle, even in the usual waveguide QED setups but becomes remarkable in the ultrastrong-coupling regime dueto the large Lamb shift. However, in contrast with the intuition provided by the input-output theory, such ellipticmotion does not propagate into the waveguide. We also obtain an analytic formula of the reflection/transmissioncoefficient, which is asymmetric with respect to the renormalized cavity frequency. We hope that the rigorous opticalresponse presented here would be useful for developing theoretical tools applicable to highly dissipative cavity andwaveguide QEDs.
II. THEORETICAL MODELA. Hamiltonian
In a setup considered in this study (Fig. 1), a cavity is coupled to a semi-infinite waveguide and a monochromaticdrive field is applied through this waveguide. In the natural units of ~ = v = 1, where v is the photon velocity in thewaveguide, the Hamiltonian of the overall system is given byˆ H = ω b ˆ b † ˆ b + Z ∞ dk h k ˆ c † k ˆ c k + ξ k (ˆ b † + ˆ b )(ˆ c † k + ˆ c k ) i , (1)where ω b is the bare cavity frequency, and ˆ b and ˆ c k are the annihilation operators of the cavity mode and thewaveguide mode with wave number k , respectively, satisfying the bosonic commutation relations, [ˆ b, ˆ b † ] = 1 and[ˆ c k , ˆ c † k ′ ] = δ ( k − k ′ ). The cavity-waveguide coupling ξ k is a real function of k . In this study, in order that the Fanodiagonalization is applicable, we assume the following conditions on ξ k [37]: (i) ξ k is nonzero for k >
0, (ii) ξ k is anodd function of k , namely, ξ − k = − ξ k , and (iii) the coupling is weak enough to satisfy Z ∞ dk ξ k /k < ω b / . (2) B. Drude-form coupling
To be more concrete, we employ a Drude-form for the cavity-waveguide coupling, ξ k = C kk + ω x , (3)where C is a constant and ω x is the cutoff frequency. We assume ω x ≫ ω b so that the coupling is Ohmic ( ∝ k ) nearthe cavity resonance. We set ω x = 5 ω b hereafter. We denote the radiative decay rate of the cavity mode into thewaveguide by κ . By naively applying the Fermi golden rule, we obtain κ = 2 πξ ω b . Therefore, we set the constant C as C = κ ( ω b + ω x )2 πω b . (4)In this paper, we employ a dimensionless quantity κ/ω b as a measure of the strength of the cavity-waveguide coupling. C. Renormalization of frequency and decay rate
Since the Fermi golden rule is in principle valid only for a weak cavity-waveguide coupling, κ may deviate from theactual decay rate e κ , particularly for a stronger coupling. Furthermore, the resonance frequency ω b also acquires aLamb shift and takes a renormalized value e ω b . As we observe later in Sec. III B, e ω b and e κ are identified as e ω b = Re( λ ) , (5) e κ = 2 Im( λ ) , (6)where λ is a complex cavity frequency, which is a solution of the cubic equation (23) in the first quadrant (Fig. 3). w b /w b ~ k /k ~ k /w b ~ ~ k /w b k / k , w b / w b , k / w b ~~~~ FIG. 2: Dependences of the cavity decay rate ( e κ/κ , solid) and resonance frequency ( e ω b /ω b , dashed) on the cavity-waveguidecoupling, κ/ω b . Their ratio, e κ/ e ω b , is plotted by a dotted line. The ultrastrong coupling ( e κ/ e ω b > .
1) is attained for κ/ω b > . e κ/ e ω b >
1) is attained for κ/ω b > . From Eq. (2), we have κ/ω b < ω b ω x / ( ω b + ω x ). This inequality sets an upper bound for the coupling strength: κ/ω b < .
192 for ω x = 5 ω b . However, as we discuss in Sec. III B, from the condition that the renormalized frequency e ω b is positive, we have a more strict upper bound, κ/ω b < . e ω b and e κ on κ/ω b . We observe that, beyond the perturbative regime of κ/ω b ≪
1, the agreement between e κ and κ is fairly good even for stronger coupling. In contrast, the renormalizedcavity frequency decreases drastically as the coupling becomes stronger. As a result, not only the ultrastrong couplingregime ( e κ/ e ω b > .
1) but also the deep-strong coupling regime ( e κ/ e ω b >
1) is attainable within this theoretical model.
D. Initial state vector
In this study, we investigate the optical response of a cavity driven by a monochromatic classical field appliedthrough the waveguide (Fig. 1). The positively rotating part of drive amplitude is given by E ( r, t ) = E d e ik d ( r − t ) , (7)where E d and k d are the complex amplitude and wavenumber/frequency of the drive, respectively. At the initialmoment ( t = 0), we assume that the whole system is in the ground state expect the drive field in the waveguide,which is in a coherent state. The initial state vector is then written as | ψ i i = exp (cid:16) √ πE d ˆ c † k d − √ πE ∗ d ˆ c k d (cid:17) | vac i , (8)where | vac i is the overall ground state.The real-space representation e c r of the waveguide field operator is defined as the Fourier transform of ˆ c k , e c r = 1 √ π Z ∞ dk e ikr ˆ c k . (9)We can check that h e c r (0) i ≡ h ψ i | e c r (0) | ψ i i = E ( r, r = 0. For example, for a closed boundary condition,the waveguide mode function takes the form of f k ( r ) = p /π sin( kr ) = ( ie − ikr − ie ikr ) / √ π [39]. Therefore, weshould add a phase factor i ( − i ) for the input (output) port in Eq. (9), which accounts for the sign flip upon reflectionat a mirror. However, we employ Eq. (9) as the real-space representation of waveguide modes for simplicity. Thisintroduces no problem except for definition of the relative phase in the input and output ports. III. DIAGONALIZATIONA. General formula
The Hamiltonian [Eq. (1)] is bilinear in bosonic operators and can be diagonalized by the Fano’s method. Whenthe cavity-waveguide coupling is weak enough to satisfy Eq. (2), we can rewrite the Hamiltonian asˆ H = Z ∞ dk k ˆ d † k ˆ d k , (10)where ˆ d k is an eigenmode annihilation operator satisfying the bosonic commutation relation,[ ˆ d k , ˆ d † k ′ ] = δ ( k − k ′ ) . (11)ˆ d k is given by linear combination of the original bosonic operators asˆ d k = β ( k )ˆ b + β ( k )ˆ b † + Z ∞ dq (cid:2) γ ( k, q )ˆ c q + γ ( k, q )ˆ c † q (cid:3) , (12)where the coefficients are given by (see Appendix A for derivation) β ( k ) = ( k + ω b ) ξ k k − ω b z ( k ) , (13) β ( k ) = ( k − ω b ) ξ k k − ω b z ( k ) , (14) γ ( k, q ) = δ ( k − q ) + e γ ( k, q ) , (15) γ ( k, q ) = 2 ω b ξ k ξ q ( k + q )[ k − ω b z ( k )] , (16)where e γ ( k, q ) = 2 ω b ξ k ξ q ( k − q − i k − ω b z ( k )] , (17)and z ( k ) is a dimensionless quantity representing the self-energy correction for the resonator frequency, z ( k ) = 1 + 2 ω b Z ∞−∞ dq ξ q k − q − i . (18)Inversely, the bare operators ˆ b and ˆ c k are expressed in terms of the eigenoperators byˆ b = Z ∞ dq [ β ∗ ( q ) ˆ d q − β ( q ) ˆ d † q ] , (19)ˆ c k = Z ∞ dq [ γ ∗ ( q, k ) ˆ d q − γ ( q, k ) ˆ d † q ] . (20) B. Specific results for Drude-form coupling
When the cavity-waveguide coupling takes the Drude form [Eqs. (3) and (4)], z ( k ) and k − ω b z ( k ) are rewrittenas follows, z ( k ) = 1 + 2 πiCω b ( k − iω x ) , (21) k − ω b z ( k ) = ( k − λ )( k − λ )( k − λ ) k − iω x , (22)where λ , , are the solutions of the following cubic equation for k , k − iω x k − ω b k + ( iω x ω b − iπCω b ) = 0 . (23) ReIm w b -w b i w x l l l FIG. 3: λ , , on the complex plane. Arrows indicate the directions as the cavity-waveguide coupling κ is increased. As shown in Fig. 3, λ ( λ ) is on the first (seond) quadrant and λ is on the positive imaginary axis. The real andimaginary parts of λ correspond to the Lamb-shifted resonance frequency e ω b and half of the decay rate e κ/ κ . The zeroth-order solutions are λ (0)1 = ω b , λ (0)2 = − ω b , and λ (0)3 = iω x . Up to the first order in κ , the three solutionsare given by λ ≈ ( ω b − κω x / ω b ) + iκ/ λ ≈ − ( ω b − κω x / ω b ) + iκ/
2, and λ ≈ iω x − iκ .For an extremely strong coupling, λ and λ also become purely imaginary. The condition that the renormalizedfrequency e ω b remain positive, in other words, λ and λ are not purely imaginary, is that κ < [ ω b ω x − f ( µ − )] / ( ω b + ω x ),where f ( x ) = x − ω x x + ω b x and µ − is a smaller root of the df /dx = 0, namely, µ − = ( ω x − p ω x − ω b ) /
3. For ω x = 5 ω b , this condition is κ/ω b < . IV. OPTICAL RESPONSEA. Cavity Amplitude
In this section, we investigate time evolution of the whole system from the initial state vector, Eq. (8). We firstobserve the amplitude of the cavity mode, h ˆ b ( t ) i ≡ h ψ i | ˆ b ( t ) | ψ i i . Since ˆ d k is an eigenoperator of the Hamiltonian, ˆ b ( t )is given, from Eq. (19), by ˆ b ( t ) = Z ∞ dq h e − iqt β ∗ ( q ) ˆ d q − e iqt β ( q ) ˆ d † q i . (24)Furthermore, | ψ i i is an eigenstate of ˆ d q and satisfiesˆ d q | ψ i i = √ π [ E d γ ( q, k d ) + E ∗ d γ ( q, k d )] | ψ i i . (25)From these results, h ˆ b ( t ) i is given by h ˆ b ( t ) i = √ πE d Z ∞ dq (cid:2) e − iqt β ∗ ( q ) γ ( q, k d ) − e iqt β ( q ) γ ∗ ( q, k d ) (cid:3) + √ πE ∗ d Z ∞ dq (cid:2) e − iqt β ∗ ( q ) γ ( q, k d ) − e iqt β ( q ) γ ∗ ( q, k d ) (cid:3) . (26)This is divided into stationary and transient components as h ˆ b ( t ) i = h ˆ b ( t ) i s + h ˆ b ( t ) i t . The stationary component isgiven by h ˆ b ( t ) i s = √ πβ ∗ ( k d ) E d e − ik d t − √ πβ ( k d ) E ∗ d e ik d t . (27)The transient component is presented in Appendix B. Putting E d = | E d | e iθ d , we haveRe h ˆ b ( t ) i s = √ π | E d | ω b ξ k d Re (cid:18) e i ( k d t − θ d ) k d − ω b z ( k d ) (cid:19) , (28)Im h ˆ b ( t ) i s = −√ π | E d | k d ξ k d Im (cid:18) e i ( k d t − θ d ) k d − ω b z ( k d ) (cid:19) . (29) k d /w b Long / s ho r t a x i a l r ad ii Im (c) k d /w b (d) -4-2 0 2 4 -4 -2 0 2 4 Re 〈 b 〉 I m 〈 b 〉 k d / w b =1.000 k d / w b =0.975Fluctuation (a) -4-2 0 2 4 -4 -2 0 2 4 Re 〈 b 〉 I m 〈 b 〉 (b) Fluctuation k d / w b =1.000 k d / w b =0.476 FIG. 4: Elliptical motion of the cavity amplitude. (a) Trajectories on the phase space for κ/ω b = 0 .
01. The drive frequencyis set at the renormalized resonance e ω b (= 0 . ω b ) (solid) and the bare resonance ω b (dashed). The photon rate of the drivefield is set at | E d | = 2 . κ , at which the mean intra-cavity photon number is estimated to be h ˆ b † ˆ b i = 4 | E d | /κ = 10 onresonance, following the input-output theory. The uncertainty ellipse is also shown. (b) The same plot as (a) for κ/ω b = 0 . e ω b = 0 . ω b . (c) Dependence of the long (solid line) and short (dotted line) axial radii on thedrive frequency k d for κ/ω b = 0 .
01. (d) The same plot as (c) for κ/ω b = 0 . These equations indicate that the motion of the cavity amplitude h ˆ b ( t ) i s on the phase space is elliptical in general;the ratio of the vertical (imaginary) radius relative to the horizontal (real) radius is k d /ω b , and thus depends on thedrive frequency. However, such elliptical motion is not remarkable when the cavity-waveguide coupling κ is small.For a small κ case, strong optical response is obtained within a narrow frequency region around the renormalizedcavity frequency e ω b , which is close to the bare frequency ω b . For example, when κ/ω b = 0 .
01, the renormalizedfrequency amounts to e ω b = 0 . ω b [Eq. (44)]. Therefore, the motion is almost circular for small κ , as we observein Figs. 4 (a) and (c). In contrast, for a large κ case, the motion on the phase space becomes highly elliptical, as weobserve in Figs. 4 (b) and (d). This is due to the large frequency renormalization (Lamb shift). When κ/ω b = 0 . e ω b = 0 . ω b . B. Quadrature Fluctuations
Here, we investigate the quadrature fluctuations of the cavity mode. We define the ˆ X and ˆ Y quadratures byˆ X = (ˆ b + ˆ b † ) / Y = − i (ˆ b − ˆ b † ) /
2, respectively, and their fluctuations by ∆ X = q h ˆ X i − h ˆ X i and ∆ Y = k /w b D X D Y( D X D Y) D X , D Y FIG. 5: Quadrature fluctuations: ∆ X (solid), ∆ Y (dashed), and √ ∆ X ∆ Y (thin dotted). ω x /ω b = 5. q h ˆ Y i − h ˆ Y i , respectively, where h ˆ O i = h ψ i | ˆ O | ψ i i . From these definitions, we have∆ X = q h ˆ b † ( t ) , ˆ b ( t ) i + 2Re h ˆ b ( t ) , ˆ b ( t ) i , (30)∆ Y = q h ˆ b † ( t ) , ˆ b ( t ) i − h ˆ b ( t ) , ˆ b ( t ) i , (31)where h ˆ O, ˆ O ′ i ≡ h ˆ O ˆ O ′ i − h ˆ O ih ˆ O ′ i . From Eqs. (24) and (25), we can confirm that both h ˆ b † ( t ) , ˆ b ( t ) i and h ˆ b ( t ) , ˆ b ( t ) i reduces to the following time-independent quantities, h ˆ b † , ˆ b i = Z ∞ dq | β ( q ) | , (32) h ˆ b, ˆ b i = − Z ∞ dqβ ∗ ( q ) β ( q ) , (33)and that the quadrature fluctuations, ∆ X and ∆ Y , are identical to those of the vacuum fluctuations. The integralsappearing in Eqs. (32) and (33) can be performed analytically for the Drude-form coupling (Appendix C).Figure 5 plots the dependences of ∆ X and ∆ Y on the cavity-waveguide coupling κ . We observe that there existssqueezing in Y quadrature, and the degree of squeezing increases for larger κ . The state is not a minimum uncertaintystate, since √ ∆ X ∆ Y > / C. Amplitude of waveguide field
From Eqs. (20) and (25), the amplitude of the waveguide field in the wavenumber representation is given by h ˆ c k ( t ) i = √ πE d Z ∞ dq (cid:2) e − iqt γ ∗ ( q, k ) γ ( q, k d ) − e iqt γ ( q, k ) γ ∗ ( q, k d ) (cid:3) + √ πE ∗ d Z ∞ dq (cid:2) e − iqt γ ∗ ( q, k ) γ ( q, k d ) − e iqt γ ( q, k ) γ ∗ ( q, k d ) (cid:3) . (34)Using Eqs. (15)–(17), this quantity is rewritten as follows, h ˆ c k ( t ) i = √ πE d (cid:2) e − ik d t δ ( k − k d ) + e − ikt e γ ( k, k d ) + e − ik d t e γ ∗ ( k d , k ) (cid:3) − i p /πω b ξ k ξ k d E d Z ∞−∞ dq e − iqt ( q − k + i q − k d − i (cid:18) q − ω b z ( q ) − q − ω b z ∗ ( q ) (cid:19) + √ πE ∗ d (cid:2) e − ikt γ ( k, k d ) − e ik d t γ ( k d , k ) (cid:3) + i p /πω b ξ k ξ k d E ∗ d Z ∞−∞ dq e iqt ( q + k − i q − k d + i (cid:18) q − ω b z ( q ) − q − ω b z ∗ ( q ) (cid:19) . (35) r (units of w b - ) no r m a li z ed a m p li t ude (a) -0.5 0 0.5 1 1.5 0 100 200 300 (b) r (units of w b - ) FIG. 6: Normalized amplitude of the waveguide field, h e c r ( t ) i /E ( r, t ). (a) Real and (b) imaginary parts. Solid lines representthe rigorous numerical results, and dotted lines represent the approximate one given by Eq. (42). The parameters are chosenas follows: κ/ω b = 0 . k d /ω b = 0 .
6, and t = 300 /ω b . The integral in the second line in the above equation can be performed by employing the residue theorem. Theintegrand has four poles in the lower complex plane of q at k − i λ ∗ , λ ∗ , and λ ∗ , and the latter three poles yieldtransient components. Therefore, the stationary component of the second line comes from the pole at k − i −√ πω b ξ k ξ k d E d e − ikt k − k d − i ( k − ω b z ( k ) − k − ω b z ∗ ( k ) ). Repeating the same arguments, the stationary componentof the fourth line of Eq. (35) is given by −√ πω b ξ k ξ k d E ∗ d e − ikt k + k d ( k − ω b z ( k ) − k − ω b z ∗ ( k ) ). As a result, the stationarycomponent of the waveguide amplitude is written as h c k ( t ) i = h c k ( t ) i (1) + h c k ( t ) i (2) + h c k ( t ) i (3) , (36) h c k ( t ) i (1) = √ πδ ( k − k d ) E d e − ik d t , (37) h c k ( t ) i (2) = √ πω b ξ k ξ k d E d k − k d − i (cid:18) e − ikt k − ω b z ∗ ( k ) − e − ik d t k d − ω b z ∗ ( k d ) (cid:19) , (38) h c k ( t ) i (3) = √ πω b ξ k ξ k d E ∗ d k + k d (cid:18) e − ikt k − ω b z ∗ ( k ) − e ik d t k d − ω b z ( k d ) (cid:19) . (39)We switch to the real-space representation, h e c r ( t ) i , using Eq. (9). h e c r ( t ) i (1) is immediately given by h e c r ( t ) i (1) = E d e ik d ( r − t ) . (40)Obviously, this is nothing but the input drive field of Eq. (7). Regarding h e c r ( t ) i (2) , the principal contribution comesfrom the pole at k = k d + i h c k ( t ) i (2) s ≈ √ πω b ξ k d E d [ k d − ω b z ∗ ( k d )] − [ k − k d − i − (cid:0) e − ikt − e − ik d t (cid:1) . Then, we have h e c r ( t ) i (2) ≈ − πiω b ξ k d k d − ω b z ∗ ( k d ) θ ( r ) θ ( t − r ) E d e ik d ( r − t ) , (41)where θ is the Heaviside step function. This represents the radiation from the cavity emitted into the positive r region. Finally, h e c r ( t ) i (3) yields no propagating wave. Combining these results, we obtain the following analytic formof h e c r ( t ) i : h e c r ( t ) i ≈ − πiω b ξ k d k d − ω b z ∗ ( k d ) θ ( r ) θ ( t − r ) ! × E d e ik d ( r − t ) . (42)The spatial shape of h e c r ( t ) i is plotted in Fig. 6, in which the rigorous shape [numerical Fourier transform of Eq. (36)]is plotted by solid lines and the approximate form [Eq. (42)] is plotted by dotted lines. We observe good agreementbetween them, except the deviations at the wavefront of the cavity radiation ( r . t ) and at the cavity position ( r ∼ κ − , which agrees with our observation in Fig. 6. On the other k d /w b a r g ( R ) ( un i t s o f p ) FIG. 7: Phase shift upon reflection as a function of the drive frequency. The cavity-waveguide coupling strength, κ/ω b , isindicated. hand, the latter deviation around the cavity position originates in the fact that the cavity-waveguide interaction hasa finite bandwidth in the wavenumber space and therefore is not spatially local in the present theoretical model. Thebandwidth of the cavity-waveguide coupling is of the order of ω b in the wavenumber space, and is therefore of theorder of ω − b in the real space. This explains the deviation localized at the origin in Fig. 6.A notable fact is that, in contrast with the intracavity field amplitude [Eq. (27)] that is composed of both positivelyand negatively oscillating components, the waveguide field amplitude in the output port [Eq. (42)] is composed onlyof the positively oscillating one. Therefore, the elliptic motion is specific to the intracavity amplitude. D. Refection coefficient
The refection coefficient is identified as R = h e c r ( t ) i /E ( r, t ) at the output port ( r > R is identifiedas R ( k d ) = 1 − πiω b ξ k d k d − ω b z ∗ ( k d ) . (43)We can check that | R | = 1 for any input frequency k d . This implies that input field is reflected completely coherently,which is characteristic to linear optical response. In Fig. 7, we plot the phase shift upon reflection, arg R , as afunction of the drive frequency k d , varying the cavity-waveguide coupling. As we increase the coupling, we observethe broadening of the linewidth and the redshift of the resonance frequency. The spectrum takes a kink-shapedform around the renormalized frequency. For a weak coupling, the spectrum is anti-symmetric with respect to therenormalized frequency, as is predicted by standard input-output theory. However, for a stronger coupling, suchsymmetry is gradually lost.We can determine the renormalized resonance frequency e ω b as the drive frequency achieving the π phase shift, R ( e ω b ) = −
1. From this condition, e ω b is analytically given by e ω b = ω b − ω x + p ( ω b + ω x )( ω b + ω x − κω x )2 . (44)As we can confirm in Fig. 2, this is almost identical to the former definition of e ω b by Eq. (5). We observe in Fig. 7 thatthe reflection coefficient becomes independent of the coupling strength κ/ω b at the bare cavity resonance, k d = ω b ;we can check that R ( ω b ) = ( ω x − iω b ) / ( ω x + iω b ). E. Open waveguide
In the previous subsection, we have determined the reflection coefficient R when a semi-infinite waveguide is coupledto a cavity (Fig. 1). From this result, we can readily determine the reflection and transmission coefficients R ′ and T ′ ,when the cavity is coupled to an open waveguide [Fig. 8(a)]. The amplitude of waveguide field in this case is written0 T r an s m i ss i v i t y (b) k d /w b rr = cavitydrive field tranmissionreflection (a) FIG. 8: (a) Schematic of a cavity coupled to an open waveguide. (b) Transmissiveity | T ′ | as a function of the drive frequency.The cavity-waveguide coupling strength, κ/ω b , is indicated. as E ( r, t ) = E d e − iω d t × ( e ik d r + R ′ e − ik d r ( r < T ′ e ik d r (0 < r ) . (45)We divide this field into even and odd components. The even component interacts with the cavity whereas the oddcomponent does not. The even component is defined by E s ( r, t ) = [ E ( r, t ) + E ( − r, t )] / E s ( r, t ) = E d e − ik d ( r + t ) + R ′ + T ′ E d e ik d ( r − t ) for r >
0. Since the first (second) term in the right-hand-side of thisequation represents the incoming (outgoing) field, we have R ′ + T ′ = R . Similarly, the odd component is defined by E a ( r, t ) = [ E ( r, t ) − E ( − r, t )] / E s ( r, t ) = − E d e − ik d ( r + t ) + T ′ − R ′ E d e ik d ( r − t ) for r > T ′ − R ′ = 1. Therefore, R ′ = ( R − / , (46) T ′ = ( R + 1) / . (47)We can readily confirm that | R ′ | + | T ′ | = 1. The transmissivity | T ′ | is plotted in Fig. 8(b) as a function of the drivefrequency. We observe that the symmetric transmission dip for a weak coupling case (solid line) gradually becomesasymmetric as the cavity-waveguide coupling increases (dashed and dotted lines). V. SUMMARY
In this study, we investigated optical response of a linear waveguide QED system, namely, an optical cavity coupledto a waveguide. Our analysis is based on exact diagonalization of the overall Hamiltonian, and is therefore rigorouseven in the ultrastrong and deep-strong coupling regimes of waveguide QED, in which the perturbative treatmentsof dissipation such as the Lindblad master equation are no longer valid. We observed that the motion of the cavityamplitude in the phase space is elliptical in general, owing to the counter-rotating terms in the cavity-waveguidecoupling. Such elliptical motion becomes remarkable in the ultrastrong coupling regime due to the large Lamb shift ofthe cavity frequency comparable to its bare frequency. However, such an elliptical motion of the cavity amplitude is notreflected in the output field, contrary to the intuition by the input-output theory. We obtained an analytic expressionof the reflection/transmission coefficient, which becomes asymmetric with respect to the resonance frequency as thecavity-waveguide coupling is increased.
Acknowledgments
The author acknowledges fruitful discussions with T. Shitara and I. Iakoupov. This work is supported in partby JST CREST (Grant No. JPMJCR1775), JST ERATO (Grant no. JPMJER1601), MEXT Q-LEAP, and JSPSKAKENHI (Grant No. 19K03684).1
Appendix A: Fano diagonalization
From Eqs. (10) and (11), we have [ ˆ d k , ˆ H ] = k ˆ d k . This leads the following equations:( k − ω b ) β ( k ) = Z ∞ dqξ q [ γ ( k, q ) − γ ( k, q )] , (A1)( k + ω b ) β ( k ) = Z ∞ dqξ q [ γ ( k, q ) − γ ( k, q )] , (A2)( k − q ) γ ( k, q ) = ξ q [ β ( k ) − β ( k )] , (A3)( k + q ) γ ( k, q ) = ξ q [ β ( k ) − β ( k )] . (A4)From Eqs. (A2) and (A4), we obtain β ( k ) = k − ω b k + ω b β ( k ) and γ ( k, q ) = k − qk + q γ ( k, q ). Then, Eqs. (A1) and (A3) arerewritten as ( k − ω b ) β ( k ) = 2 Z ∞ dq qξ q k + q γ ( k, q ) , (A5)( k − q ) γ ( k, q ) = 2 ω b k + ω b β ( k ) ξ q . (A6)Equation (A6) is rewritten as γ ( k, q ) = 2 ω b k + ω b β ( k ) ξ q (cid:18) k − q − i y ( k ) δ ( k − q ) (cid:19) , (A7)where y ( k ) is a quantity to be determined. Substituting the above equation into Eq. (A5), and using R ∞ qξ q ( k + q )( k − q − i = R ∞−∞ ξ q k − q − i , y ( k ) is given by y ( k ) = 1 ξ k (cid:18) k − ω b ω b − Σ( k ) (cid:19) , (A8)Σ( k ) = Z ∞−∞ dq ξ q k − q − i . (A9)Note that Σ( k ) is the self-energy of the cavity, satisfying Σ( − k ) = Σ ∗ ( k ) and ImΣ( k ) = πξ k .Up to here, we derived the expressions of β , γ and γ in terms of β . β ( k ) is determined by the normalization con-dition, Eq. (11). This is rewritten as δ ( k − k ′ ) = β ( k ) β ∗ ( k ′ ) − β ( k ) β ∗ ( k ′ )+ R ∞ dq [ γ ( k, q ) γ ∗ ( k ′ , q ) − γ ( k, q ) γ ∗ ( k ′ , q )],which leads to ω b ξ k ( k + ω b ) | β ( k ) || y ( k ) | = 1. By adequately choosing the phase of β , we obtain Eq. (13), β ( k ) = k + ω b ω b ξ k y ( k ) = ( k + ω b ) ξ k k − ω b z ( k ) . (A10) β , γ and γ are obtained accordingly. Appendix B: transient component of cavity mode
Here we present the transient component of the cavity amplitude, h ˆ b ( t ) i t , which is omitted in Sec. IV A: h ˆ b ( t ) i t = √ πE d ω b ξ k d Z ∞−∞ dq e − iqt ( q + ω b ) ξ q ( q − k d − i q − ω b z ( q )][ q − ω b z ∗ ( q )] − √ πE ∗ d ω b ξ k d Z ∞−∞ dq e iqt ( q − ω b ) ξ q ( q − k d + i q − ω b z ( q )][ q − ω b z ∗ ( q )] . (B1)Using ω b ξ q [ q − ω b z ( q )][ q − ω b z ∗ ( q )] = iπ ( q − ω b z ( q ) − q − ω b z ∗ ( q ) ) and that q − ω b z ( q ) has no poles on the lower half plane,transient component is rewritten as h b ( t ) i t = iE d ξ k d √ π Z ∞−∞ dq e − iqt ( q + ω b )( q − k d − i q − ω b z ∗ ( q )] + iE ∗ d ξ k d √ π Z ∞−∞ dq e iqt ( q − ω b )( q − k d + i q − ω b z ( q )] . (B2)2 Appendix C: Integrals in Eqs. (32) and (33)
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