Oscillating bound states for a giant atom

We investigate the relaxation dynamics of a single artificial atom interacting, via multiple coupling points, with a continuum of bosonic modes (photons or phonons) in a one-dimensional waveguide. In the non-Markovian regime, where the travelling time of a photon or phonon between the coupling points is sufficiently large compared to the inverse of the bare relaxation rate of the atom, we find that a boson can be trapped and form a stable bound state. More interestingly, if the number of coupling points is more than two, the bound state can oscillate persistently by exchanging energy with the atom despite the presence of the dissipative environment. We propose several realistic experimental schemes to generate such oscillating bound states.

We investigate the relaxation dynamics of a single artificial atom interacting, via multiple coupling points, with a continuum of bosonic modes (photons or phonons) in a one-dimensional waveguide. In the non-Markovian regime, where the travelling time of a photon or phonon between the coupling points is sufficiently large compared to the inverse of the bare relaxation rate of the atom, we find that a boson can be trapped and form a stable bound state. More interestingly, if the number of coupling points is more than two, the bound state can oscillate persistently by exchanging energy with the atom despite the presence of the dissipative environment. We propose several realistic experimental schemes to generate such oscillating bound states.
Introduction.-The study of interaction between light and matter is one of the core topics in modern physics [1]. In such studies, the wavelength of the light is usually large compared to the size of the (artificial) atoms constituting the matter [2][3][4][5][6][7]. Indeed, the traditional framework of quantum optics is based on point-like atoms [8] and neglects the time it takes for light to pass a single atom. Recently, following significant technological advances for superconducting circuits [7,[9][10][11], "giant" artificial atoms (transmon qubits [12]) have been designed to interact with surface acoustic waves (SAWs) via multiple coupling points in a waveguide [13][14][15] (or resonator [16][17][18][19][20][21][22]) as sketched in Fig. 1 (left inset). Such a giant-atom structure can also be realised in a more conventional circuit-quantum-electrodynamics (circuit-QED) experiment by coupling a single Xmon [23], a version of the transmon, to a meandering coplanar waveguide (CPW) as sketched in Fig. 1 (right inset) [24][25][26]. Since the distance between coupling points can be (much) longer than the characteristic wavelength of the bath, it is necessary to consider the phase difference between these coupling points. Striking effects have been found as a consequence of this, e.g., frequency-dependent relaxation rate and Lamb shift of a giant atom [24][25][26], and decoherence-free interaction between multiple giant atoms [25,27]. The giant-atom scheme has recently been extended to higher dimensions with cold atoms [28] and constitutes an exciting new paradigm in quantum optics [10,28], where much remains to explore.
In this Letter, we investigate the relaxation dynamics of a single giant atom interacting with a one-dimensional (1D) bosonic bath (e.g., an open waveguide for phonons or photons) through multiple coupling points. Our main result is that three or more coupling points enable the creation of persistently oscillating bound states, a phenomenon which, to the best of our knowledge, is unique to giant atoms. We envision that this phenomenon could be used in quantum information processing as a singlephoton (-phonon) "tweezer" or trap, and that it could be viewed as a minimalistic implementation of cavity QED with the atom forming its own cavity.
Model Hamiltonian.-We consider a two-level atom interacting with an open 1D waveguide at N coupling points |Fig. 1 illustrates the case N = 3]. As illustrated by the two insets in Fig. 1, this system can be implemented in at least two different experimental schemes: a transmon qubit with multiple interdigital transducers (IDTs) coupled to SAWs through piezoelectric effects [13][14][15][16] or an Xmon qubit [23][24][25][26] with multiple arms capacitively coupled to a coplanar waveguide. The total Hamiltonian for the system is where we have defined the atomic operators σ + = |e g| and σ − = (σ + ) † with |g (|e ) the atomic ground (excited) state and Ω the atomic transition frequency. The parameters k, v, and ω k = |k|v are the wave vectors, velocities, and frequencies of the bosonic fields (phonons or photons) in the waveguide. The field operators a k satisfy a k , a † k = δ(k − k ). The rotating-wave approximation (RWA) has been applied in the interaction term. We assume a constant effective relaxation rate γ at each coupling point, located at x m (m = 1, 2, · · · , N). We also assume the coupling points are equidistant. Thus, the travel time for bosons between two neighbouring coupling points is a constant τ = (x m+1 − x m )/v. In this work, we investigate novel phenomena arising from non-Markovian dynamics due to τ being non-negligible.
Equations of motion and their solutions.-We study the process of spontaneous emission from the giant atom into the waveguide. The atom begins in the excited state |e and the field in the waveguide is in the vacuum state |vac . Since the total number of atomic and field excitations is conserved in Eq. (1) due to the RWA, we study the single-excitation subspace of the full system. The total system state can thus be described by where the integral describes the state of a single boson propagating in the waveguide. From the Schrödinger equation i ∂/∂t|Ψ (t) = H|Ψ (t) , following the method in Ref. [57], we derive the equation of motion (EOM) for the probability amplitude of the giant atom being excited, and the time evolution of the bosonic field function Here, Θ(•) is the Heaviside step function, which describes time-delayed feedback among the coupling points. The field intensity function p(x, t) ≡ |ϕ(x, t)| 2 describes the probability density at position x and time t to find a single phonon or photon for all possible wave vectors k.
The first term on the right-hand side of Eq. (3) describes the coherent dynamics of the atom. The second and third terms describe the relaxation processes due to Markovian and non-Markovian dynamics, respectively. The solution of β(t) can be obtained by a Laplace transformation: where the complex frequency parameters s n are given by the solutions to the equation For finite time delay τ > 0, the nonlinear Eq. (6) has multiple solutions. In general, there is no simple closed form for these solutions.
Dark-state condition.-Usually, the complex frequency s n has a negative real part, which represents the relaxation rate. In some particular situations, s n can be purely imaginary. In that case, the corresponding mode is a dark state, which does not decay despite the dissipative environment. We seek the purely imaginary solution s n ≡ −iΩ n with Plugging this into Eq. (6), we obtain the following condition for the dark states: Note that for the RWA to hold, we require |Ω n − Ω|/Ω 1 or, equivalently, Nγ 2Ω cot nπ N 1 and n ∈ Z + according to Eq. (8). In the Markov limit γτ → 0, the dark-state condition Eq. (8) is simplified into Ωτ = 2nπ/N and the dark frequency is Ω n = Ω + 1 2 Nγ cot nπ N [24]. In the non-Markovian limit of sufficiently large γτ, the additional nonlinear cotangent term in Eq. (8) cannot be neglected. Due to this term, there is an associated bound field state in the waveguide for a given dark state of the atom.
From Eqs. (4) and (9), we calculate [59] the explicit expression for the field density in the long-time limit, p n (x) ≡ p(x, t → ∞), for a given dark state s n : Here, we have relabelled the position coordinate by x = (m − 1 + λ)vτ with m = 1, 2, . . . , N and λ ∈ [0, 1). Equation (10) is only valid for the position between the two outermost coupling points, i.e., We calculate [59] the total field intensity I(n) of the bound field state for a given dark state: (11) We see that, in the Markovian limit γτ → 0, the total field strength I(n) → 0. Thus, the bound state only exists in the non-Markovian regime, where γτ is sufficiently large. In the special case of N = 2, the dark-state condition Eq. (8) can only be fulfilled for odd integers n, and the residual field strength is I(n) = γτ/(1 + γτ) 2 ≤ 1/4. In Oscillating bound states.-The dark-state condition Eq. (8) is a nonlinear equation for integer n and γτ > 0. It is possible to find two integers n 1 and n 2 satisfying Eq. (8) simultaneously. This means that, in the longtime limit after all the dissipative modes die out, the dynamics of the atomic excitation probability amplitude β(t) is a superposition of two dark states with different frequencies Ω n 1 and Ω n 2 . As a result, the atomic excitation probability |β(t)| 2 oscillates persistently with frequency Ω n 1 − Ω n 2 despite the dissipative environment. In  Fig. 3(a), we show the population dynamics for a threeleg giant atom (N = 3) with two coexisting dark states: s n=14 and s n=16 . The undamped oscillation of |β(t)| 2 indicates that the atom exchanges energy with the bosonic bath persistently. In Fig. 3(b), we plot the corresponding time evolution of the field intensity in the waveguide, showing an oscillating bound state in the long-time limit. In the Supplemental Video, we show an animation of the time evolution for the atomic excitation probabilitiy and the field intensity in the waveguide.
The amplitude of the persistent oscillations is thus A(n 1 )A(n 2 ).
(14) According to Eq. (2), the quantity |β(t)| 2 + I(n 1 , n 2 ) is the total excitation probability of the atom and the field, which is conserved, since the oscillating bound state does not decay. This gives an additional condition for the coexisting dark states: Combing this with Eq. (8), we find that the solutions are of the form n 1 = pN + n and n 2 = qN − n with p, q ∈ Z + and 1 ≤ n < N. The conditions in Eq. (12) then become Ωτ/2π = (p+q)/2 and γτ/2π = (p − q)/N + 2n/N 2 tan nπ N . By setting p ≥ q and 1 ≤ n < N/2, Eq. (12) can be satisfied and we obtain the frequencies of the two dark modes: Ω ± 1 2 Nγ cot nπ N . In Fig. 3(c), we show the existence of oscillating bound states (solid dots) in the Ωτγτ parameter space for a giant atom with N = 3. The condition in Eq. (12) implies that, if n 1 and n 2 are solutions yielding coexisting dark states, the integers n 1 + N and n 2 + N are also solutions of coexisting dark states with γτ unchanged but Ωτ increased by 2π. This results in the 2π periodicity along the horizontal direction in Fig. 3(c). The dots in the green region are beyond RWA, where the dark-mode frequency Ω n 1(2) − Ω /Ω > 0.1.
If the giant atom only has two coupling points (N = 2), the nonlinear cotangent term in condition (8) is either zero or infinity. Therefore, the oscillating bound states only exist for more than two coupling points (N ≥ 3).
Continuum limit.-We now discuss the limit of infinitely many coupling points (N → ∞). In this case, the time it takes for the field in the waveguide to pass all coupling points is Nτ → T . For capacitive coupling between the atom and the waveguide, the interaction strength g at a single point is proportional to the local capacitance c, i.e., g ∝ c [12,15] and the relaxation rate is γ ∝ g 2 ∝ c 2 [24,57]. As a result, the parameter N 2 γ ∝ (Nc) 2 , where Nc is the total capacitance, is a converged quantity N 2 γ → Γ, which describes the total relaxation rate of the atom into the waveguide. In this continuum limit, the dark-state condition Eq. (8) becomes The solution is n = (4π) −1 ΩT ± (ΩT ) 2 + 4ΓT ∈ Z, and the corresponding dark-mode frequency is Ω n = Ω + Γ 2nπ . The field intensity p n (x) of the bound state can be calculated from Eq. (10), yielding p n (x) = 2n 2 π 2 /ΓT 2n 2 π 2 /ΓT + 1 2  where L = x N − x 1 . The total field intensity of the bound state is I(n) = 3n 2 π 2 /ΓT 2n 2 π 2 /ΓT + 1 −2 ≤ 3/8. However, since the RWA condition requires n > 0, and we only have one solution fulfilling that condition, it is not expected that an oscillating bound state can be created in this case.
Note that the EOM (3) also describes the linear (classical) problem where a single harmonic mode instead of an atom interacts with the continuum of modes in an infinite waveguide. Therefore, our predictions can be immediately applied to this linear (classical) system. In Fig. 4(a), we show a continuum metal contacting capacitively with an infinite SAW waveguide made of piezoelectric material. The metal is attached to an LC circuit to tune the plasmon frequency in the metal. If the dark condition in Eq. (16) is satisfied, we expect to observe a bound state in the waveguide. To generate an oscillating bound state, we can design the contact part of the metal as a comb-like structure as shown in Fig. 4(b). Note that the two integers n 1 = N+n and n 2 = N−n with 1 ≤ n < N/2 always satisfy the dark-state condition Eq. (12). In the limit of infinitely many coupling points N n, i.e., for a very extended comb, we have Ωτ = 2π and ΓT → (2nπ) 2 . In this parameter setting, we can create two coexisting dark modes with frequencies Ω ± → Ω ± Γ 2πn . We show the field intensity of bound states in the 1D waveguide for the dark state n = 1 in Fig. 4(b) and (c).
Discussion and conclusion.-We have shown that a giant atom with N ≥ 3 coupling points to an open waveguide can harbour oscillating bound states. To observe these states in experiment, the coherence time of the (artificial) atom must exceed the oscillation period. For a transmon or Xmon qubit, the coherence time can be on the order of hundreds of microseconds [7,23,[60][61][62], which is much longer than the oscillation period shown in Fig. 3(a) since, typically, Ω/2π is several gigahertz.
One application of these bound states in quantum information processing could be as a single-photon (phonon) trap. Furthermore, the oscillating bound state, i.e., the dynamical exchange of excitations between the atom and the bosonic bound state, indicates that it is possible to realise a minimal version of cavity QED with a single giant atom in the open waveguide, reminiscent of the recent demonstration of cavity QED with atom-like mirrors [63]. In this section, we derive Eqs. (10) and (11) in the main text. For a given dark mode s n = −i 2nπ Nτ , the corresponding field intensity can also be calculated from Eqs. (4) and (9) in the main text. By parametrizing the position coordinate as x = (m − 1)vτ + λvτ with m = 1, 2, . . . , N and λ ∈ [0, 1), we have p n (x) = p(x, t → +∞) and This distribution is valid for x between x 1 and x N in the waveguide. We see that at the two ends of the giant atom, x 1 = 0 (i.e., m = 1 and λ = 0) and x N = (N − 1)vτ (i.e., m = N and λ = 0), the intensity vanishes. When the position x is outside the interval [x 1 , x m ], since the sign of (x − x m ) is fixed, the summation in the second line gives zero. This is reasonable since the excitations outside the outermost coupling points will propagate away in the waveguide and never come back.