Spectrum of single-photon emission and scattering in cavity optomechanics

Spectrum of single-photon emission and scattering in cavity optomechanics

Jie-Qiao Liao Department of Physics and Institute of Theoretical Physics, The Chinese University of Hong Kong, Shatin, Hong Kong Special Administrative Region, People’s Republic of China    H. K. Cheung Department of Physics and Institute of Theoretical Physics, The Chinese University of Hong Kong, Shatin, Hong Kong Special Administrative Region, People’s Republic of China    C. K. Law Department of Physics and Institute of Theoretical Physics, The Chinese University of Hong Kong, Shatin, Hong Kong Special Administrative Region, People’s Republic of China
July 13, 2019
Abstract

We present an analytic solution describing the quantum state of a single photon after interacting with a moving mirror in a cavity. This includes situations when the photon is initially stored in a cavity mode as well as when the photon is injected into the cavity. In addition, we obtain the spectrum of the output photon in the resolved-sideband limit, which reveals spectral features of the single-photon strong-coupling regime in this system. We also clarify the conditions under which the phonon sidebands are visible and the photon-state frequency shift can be resolved.

pacs:
42.50.Wk, 42.50.Pq, 07.10.Cm

Cavity optomechanics Kippenberg2008 (); Marquardt2009 () is a new frontier for exploring the coherent coupling via radiation pressure between electromagnetic and mechanical degrees of freedom. Several recent experimental systems in cavity optomechanics Gupta2007 (); Brennecke2008 (); Eichenfield2009 () approached the single-photon strong-coupling regime, in which the radiation pressure from a single photon can produce observable effects. Such a strong coupling is important to realize various proposals of generating macroscopic superposition states for testing quantum theory Mancini1997 (); Bose1997 (); Bouwmeester2003 (). In parallel with the progress in experiments, some theoretical investigations have also begun Rabl2011 (); Nunnenkamp2011 (); Chenp2011 () to explore the effects of radiation-pressure coupling at few- and even single-photon levels. For example, Rabl Rabl2011 (), and Nunnenkamp and coworkers Nunnenkamp2011 () have independently studied the statistics of cavity photons in the single-photon strong-coupling regime when the cavity is weakly driven by a continuous-wave laser.

Physically, in the single-photon strong-coupling regime, the mirror’s displacement induced by a single photon can significantly affect the cavity field, leading to some observable features in the spectrum of single-photon emission and scattering. Thus, a natural question is how these spectra may characterize the single-photon strong-coupling regime. In this Brief Report we answer the question by calculating analytically the spectrum of single-photon emission and scattering in a cavity optomechanical system. In particular, we indicate the relation between the photon spectral characteristic and the interaction strength [cf. Eq. (1)] of radiation pressure per photon.

To begin with, we specify the system of a Fabry-Pérot-type optomechanical cavity formed by a fixed end mirror and a moving end mirror [Fig. 1(a)]. The cavity field and the mirror motion are coupled to each other via radiation pressure. The Hamiltonian of the optomechanical cavity is

 Hopc=ℏωca†a+ℏωMb†b−ℏga†a(b†+b), (1)

where and are, respectively, the annihilation (creation) operators of the electromagnetical and mechanical modes, with the respective resonant frequencies and . The parameter is the single-photon coupling strength of the radiation pressure between the cavity and the mirror, where is the zero-point fluctuation of the mirror with mass , and is the rest length of the cavity.

Let us denote the harmonic-oscillator number states for the cavity and the mirror as and (), respectively, then the eigensystem of Hamiltonian (1) can be written as

 Hopc|m⟩a|~n(m)⟩b=ℏ(mωc+nωM−m2δ)|m⟩a|~n(m)⟩b (2)

for , where is the photon-state frequency shift caused by a single-photon radiation pressure. The state

 |~n(m)⟩b≡exp[mβ0(b†−b)]|n⟩b (3)

is an -photon displaced number state Oliveira1990 () with . Particularly, is the single-photon displaced number state; and is the zero-photon displaced number state, which is the same as the harmonic-oscillator number state by Eq. (3). For convenience, the energy-level structure of the optomechanical cavity in the zero- and one-photon cases is shown in Fig. 1(b).

In optomechanical systems, the decay rate of the mirror motion can be much smaller than the cavity-field’s decay rate , then during the time interval , the damping of the mirror motion is negligible. In the following we merely consider the dissipation of the cavity field by coupling it to continuous outside fields so that the full Hamiltonian is

 H=Hopc+∫∞0ℏωkc†kckdk+ℏξ∫∞0(c†ka+a†ck)dk, (4)

where () is the annihilation (creation) operator of the th outside-field mode with frequency , and is the photon-hopping interaction strength.

Denoting for concision, a general state in the single-photon subspace of the total system can be expressed as

 |φ(t)⟩ = ∞∑n=0An(t)|1⟩a|~n⟩b|∅⟩ (5) +∞∑n=0∫∞0Bn,k(t)|0⟩a|n⟩b|1k⟩dk,

where stands for the state with one photon in the cavity, no photon in these outside fields, and the mirror in displaced number state (hereafter we call the single-photon displaced number state as displaced number state when there is no confusion), while denotes the state with a vacuum cavity field, one photon in the th mode of these continuous fields, and the mirror in number state . In Eq. (5), we have used the completeness relation of these displaced number states (). According to the Schrödinger equation, we can obtain equations of motion [in the rotating picture with respect to ] for probability amplitudes as

 ˙Am(t)= −i(mωM−δ)Am(t) −iξ∞∑n=0∫∞0⟨~m|b|n⟩bBn,k(t)dk, (6a) ˙Bm,k(t)= −i(mωM+Δk)Bm,k(t) −iξ∞∑n=0⟨m|b|~n⟩bAn(t), (6b)

where is the detuning of the th mode photon from the cavity frequency. In the following we discuss the long-time solution and the spectrum of two cases.

Single-photon emission. In this case, we assume that a single photon is initially stored in the cavity mode, and such a photon will be transmitted (or emitted) out of the cavity. We first calculate the case where the initial state of the mirror is harmonic-oscillator number state . Once the solution for this case is found, the solution for general initial states of the mirror can be obtained accordingly by superposition. For the initial state , we have

 Am(0)=⟨~m|b|n0⟩b,Bm,k(0)=0. (7)

The transient solution of Eq. (6) can be obtained by the Laplace transform method. The results are

 An0,m(t)= ⟨~m|b|n0⟩be−[γc2+i(mωM−δ)]t, (8a) Bn0,m,k(t)= √γc2π∞∑n=0⟨m|b|~n⟩b⟨~n|b|n0⟩bΔk+δ−(n−m)ωM+iγc2 ×(e−i(mωM+Δk)t−e−[γc2+i(nωM−δ)]t), (8b)

where is the cavity-field decay rate. Notice that the subscript in and is added to mark the mirror’s initial state . In the long-time limit, we have and

 Bn0,m,k(∞)=√γc2π∞∑n=0⟨m|b|~n⟩b⟨~n|b|n0⟩be−i(mωM+Δk)tΔk+δ−(n−m)ωM+iγc2. (9)

We point out that the long-time limit here actually refers to the time scale of . During this time duration, the single photon leaks completely out of the cavity and the damping of the mechanical motion is negligible.

To understand Eq. (9), we note that the mirror’s initial state is a superposition of displaced number states . The transition can occur in the process of cavity photon leakage . For the process , the frequency of the emitted photon is governed by the resonance condition [see Fig. 1(b)]

 Δk=(n−m)ωM−δ, (10)

which is consistent with the real part of the pole in Eq. (9), [i.e., ]. The amplitude for the process is proportional to the overlap , which can be calculated with the relation Oliveira1990 ()

 ⟨m|b|~n⟩b=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩√m!n!e−β202(−β0)n−mLn−mm(β20),n≥m,√n!m!e−β202βm−n0Lm−nn(β20),m>n, (11)

where is the associated Laguerre polynomial.

A useful quantity in this system is the final reservoir occupation spectrum  Garraway2008 (), which is the probability density for finding the single photon in the th mode of these outside fields. When the mirror is initially in a pure state or a mixed state , the single-photon emission spectra are, respectively, given by

 S(Δk) =∞∑m=0∣∣ ∣∣∞∑n0=0Cn0Bn0,m,k(∞)∣∣ ∣∣2, (12a) S(Δk) =∞∑m=0∞∑n0=0Pn0∣∣Bn0,m,k(∞)∣∣2. (12b)

For a thermal equilibrium state with thermal phonon number , we have .

To observe the spectral signatures of the coupling strength , in Fig. 2 we plot the spectrum of single-photon emission as a function of the photon frequency (in rotating picture) for various values of when the mirror is initially prepared in ground state . Figure 2(a) is plotted for , there are no phonon sidebands. However, spectral peaks of phonon sidebands become visible when [Figs. 2(b) to 2(d)]. Such a condition can be understood by examining the overlap of Lorentizians in Eq. (9). To resolve a peak in the spectrum, the peak height should be higher than the tail of its neighboring Lorentizian. This requires in the resolved sideband regime remark ().

From Eq. (10) the peaks of these sidebands are located at . Therefore there is always a peak at (for ), which is shown in Figs. 2(b) to 2(d). In particular for , the peak appears at the first (labeled from right to left in the region ) peak in the red sideband. We remark that the photon-state frequency shift can be resolved from the Lorentzian spectrum when , which requires .

To illustrate how the spectrum depends on the mirror’s initial state, we plot in Fig. 3 the emission spectrum as a function of the photon frequency . Four different initial states are considered here: displaced ground state , ground state , coherent state , and thermal state . When the mirror is initially in , the photon leaves the cavity with a frequency smaller than , and so there are only peaks at the red sideband. In contrast, for the cases of , , and , the mirror in the displacement representation have some excited-state populations because of . Consequently there will be some probabilities for the single photon absorbing the phonon’s energy and leaving the cavity with a frequency larger than , (i.e., there are some peaks at the blue sideband). Figure 3 also indicates that the number of these peaks depends on the initially contributing phonon distribution in the mirror. The wider the contributing phonon distribution is, the more the peak’s number becomes.

Single-photon scattering.—Now we turn to the photon scattering problem in which a single photon is injected into the cavity. We assume that initially the cavity is in a vacuum, the single photon in a Lorentzian wave packet Liao2010 (), and the mirror in number state , that is,

 Am(0)=0,Bm,k(0)=√ϵπδm,n0Δk−Δ0+iϵ, (13)

where and are the detuning and spectral width of the photon, respectively. The scattering solution of the system is obtained as

 An0,m(t)= √ϵπ−2πξ⟨~m|b|n0⟩bϵ−γc2+i[(n0−m)ωM+Δ0+δ] ×(e−[γc2+i(mωM−δ)]t−e−[ϵ+i(n0ωM+Δ0)]t), (14a) Bn0,m,k(t)= √ϵπ1Δk−Δ0+iϵδm,n0e−i(mωM+Δk)t +∞∑n=0iγc√ϵ/π⟨m|b|~n⟩b⟨~n|b|n0⟩bϵ−γc2+i[(n0−n)ωM+Δ0+δ] ×(e−i(mωM+Δk)t−e−[γc2+i(nωM−δ)]t)γc2+i[(n−m)ωM−δ−Δk] −∞∑n=0iγc√ϵ/π⟨m|b|~n⟩b⟨~n|b|n0⟩bϵ−γc2+i[(n0−n)ωM+Δ0+δ] ×(e−i(mωM+Δk)t−e−[ϵ+i(n0ωM+Δ0)]t)ϵ+i[(n0−m)ωM+Δ0−Δk]. (14b)

In the long-time limit, and

 Bn0,m,k(∞) = √ϵπe−i(mωM+Δk)tΔk−Δ0+iϵδm,n0 (15) −√ϵπiγcΔk−[Δ0+(n0−m)ωM]+iϵ ×∞∑n=0⟨m|b|~n⟩b⟨~n|b|n0⟩be−i(mωM+Δk)tΔk+δ−(n−m)ωM+iγc2.

The first term in Eq. (15) corresponds to the process in which the photon is reflected by the fixed cavity mirror without entering the cavity. The second term in Eq. (15) comes from the interaction process after the single photon entering the cavity. Note that the summation part in the second term of Eq. (15) has the same form as that appearing in single-photon emission process discussed above.

Let us first consider the case of . We plot in Fig. 4 the spectrum of single-photon scattering versus the photon detuning for various when the mirror is initially prepared in ground state . It can be seen from Fig. 4 that phonon sidebands appear in the spectrum when . In addition, the sideband peaks in the spectrum can also be used to characterize the coupling strength through the frequency shift , similar to Fig. 2. It is interesting that these sidebands in the spectrum show both peaks and dips. The reason for these tips is quantum interference between the direct photon reflection channel and the scattering channel relating with the mirror’s final state , which are, respectively, the first and second terms in of Eq. (15).

We next consider the case in which the frequency of the incident photon is nearly monochromatic, (i.e., ). The motivation for studying the case is to see the resonant transition processes by tuning the incident photon frequency . In Figs. 5(a) and 5(b), we choose such that the incident photon will excite resonantly the transition . A subsequent cavity photon decay will induce transitions . As a result, there are only some peaks at the red sideband, and the first sideband peak measured from is located at . On the other hand, Figs. 5(b) to 5(d) provide a comparison of the scattering spectrum when the incident photon frequencies are: , , and such that the photon resonantly excites the system from states to , , and , respectively. When the photon leaks out of the cavity, the mirror will transition from , , and to , respectively. Then according to Fig. 1(b) the maximal-frequency sideband peak should be located at , , and , as illustrated in Fig. 5.

In conclusion, we have calculated analytically the spectrum of single-photon emission and scattering in a cavity optomechanical system. We have also indicated the connection between the spectral features and the interaction strength of radiation pressure per photon. In the resolved sideband regime , the phonon sidebands are visible when , while the condition for resolving the photon-state frequency shift is .

This work is partially supported by a grant from the Research Grants Council of Hong Kong, Special Administrative Region of China (Project No. CUHK401810).

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