Spontaneous decay of artificial atoms in a three-qubit system

Abstract

We study the evolution of qubits amplitudes in a one-dimensional chain consisting of three equidistantly spaced noninteracting qubits embedded in an open waveguide. The study is performed in the frame of single-excitation subspace, where the only qubit in the chain is initially excited. We show that the dynamics of qubits amplitudes crucially depend on the value of kd, where k is the wave vector and d is a distance between neighbor qubits. If kd is equal to an integer multiple of $\pi$, then the qubits are excited to a stationary level. In this case, it is the dark states which prevent qubits from decaying to zero, even though they do not contribute to the output spectrum of photon emission. For other values of kd, the excitations of qubits exhibit the damping oscillations which represent the vacuum Rabi oscillations in a three-qubit system. In this case, the output spectrum of photon radiation is determined by a subradiant state which has the lowest decay rate. We also investigated the case with the frequency of a central qubit being different from that of the edge qubits. In this case, the qubits’ decay rates can be controlled by the frequency detuning between the central and the edge qubits.

Appendices

Appendix A: Equations for qubits amplitudes in the Wigner–Weisskopf approximation

We assume that the first and the third qubits are identical $(\Omega _1 = \Omega _3 \equiv \Omega ,\;g_k^{(1)} = g_k^{(3)} \equiv g_k)$. The frequency and the coupling of the second qubit are different ($\Omega _2 \equiv \Omega _0 ,\,g_k^{(2)} \equiv g_k^{(0)}$). A distance between central qubit and the edge qubits is equal to d. We take the origin in the location of the second qubit: $x_1= -d, x_2=0, x_3=+d$. For this case, we expand the Eq. (7) as a set of three equations

$$\begin{aligned} \frac{{\mathrm{{d}}\beta _1 }}{{\mathrm{{d}}t}}&= - \sum \limits _k {g_k^2 } \int \limits _0^t {\beta _1 (t')e^{ - i(\omega _k - \Omega )(t - t')} \mathrm{{d}}t'} \nonumber \\&- \sum \limits _k {g_k^{} g_k^{(0)} e^{ - ikd} e^{i(\Omega - \Omega _0 )t} } \int \limits _0^t {\beta _2 (t')e^{ - i(\omega _k - \Omega _0 )(t - t')} \mathrm{{d}}t'} \nonumber \\&- \sum \limits _k {g_k^2 e^{ - i2kd} } \int \limits _0^t {\beta _3 (t')e^{ - i(\omega _k - \Omega )(t - t')} \mathrm{{d}}t'} \end{aligned}$$

(A1)

$$\begin{aligned} \frac{{\mathrm{{d}}\beta _2 }}{{\mathrm{{d}}t}}&= - \sum \limits _k {\left( {g_k^{(0)} } \right) ^2 } \int \limits _0^t {\beta _2 (t')e^{ - i(\omega _k - \Omega _0 )(t - t')} \mathrm{{d}}t'} \nonumber \\&- \sum \limits _k {g_k^{(0)} g_k^{} e^{ikd} e^{i(\Omega _0 - \Omega )t} } \int \limits _0^t {\beta _1 (t')e^{ - i(\omega _k - \Omega )(t - t')} \mathrm{{d}}t'} \nonumber \\&- \sum \limits _k {g_k^{(0)} g_k^{} e^{ - ikd} e^{i(\Omega _0 - \Omega )t} } \int \limits _0^t {\beta _3 (t')e^{ - i(\omega _k - \Omega )(t - t')} \mathrm{{d}}t'} \nonumber \\ \end{aligned}$$

(A2)

$$\begin{aligned} \frac{{\mathrm{{d}}\beta _3 }}{{\mathrm{{d}}t}}&= - \sum \limits _k {g_k^2 } \int \limits _0^t {\beta _3 (t')e^{ - i(\omega _k - \Omega )(t - t')} \mathrm{{d}}t'} \nonumber \\&- \sum \limits _k {g_k^2 e^{i2kd} } \int \limits _0^t {\beta _1 (t')e^{ - i(\omega _k - \Omega _{} )(t - t')} \mathrm{{d}}t'} \nonumber \\&- \sum \limits _k {g_k^{} g_k^{(0)} e^{ikd} e^{i(\Omega - \Omega _0 )t} } \int \limits _0^t {\beta _2 (t')e^{ - i(\omega _k - \Omega _0 )(t - t')} \mathrm{{d}}t'} .\nonumber \\ \end{aligned}$$

(A3)

According to Wigner–Weisskopf approach, we replace $\beta _1 (t'),\beta _2 (t'),\beta _3 (t')$ in the integrands with $\beta _1 (t),\beta _2 (t),\beta _3 (t)$ and take them out of the integrals

$$\begin{aligned} \frac{{\mathrm{{d}}\beta _1 }}{{\mathrm{{d}}t}}&= - \beta _1 (t)\sum \limits _k {g_k^2 } I_k (\Omega ,t) \nonumber \\&\quad - \beta _2 (t)\sum \limits _k {g_k^{} g_k^{(0)} e^{ - ikd} e^{i(\Omega - \Omega _0 )t} } I_k (\Omega _0 ,t) \nonumber \\&\quad - \beta _3 (t)\sum \limits _k {g_k^2 e^{ - i2kd} } I_k (\Omega ,t) , \end{aligned}$$

(A4)

$$\begin{aligned} \frac{{\mathrm{{d}}\beta _2 }}{{\mathrm{{d}}t}}&= - \beta _2 (t)\sum \limits _k {|g_k^{(0)} |^2 } I_k (\Omega _0 ,t) \nonumber \\&\quad - \beta _1 (t)\sum \limits _k {g_k^{(0)} g_k^{} e^{ikd} e^{ - i(\Omega - \Omega _0 )t} } I_k (\Omega ,t) \nonumber \\&\quad - \beta _3 (t)\sum \limits _k {g_k^{(0)} g_k^{} e^{ - ikd} e^{ - i(\Omega - \Omega _0 )t} } I_k (\Omega ,t) ,\nonumber \\ \end{aligned}$$

(A5)

$$\begin{aligned} \frac{{\mathrm{{d}}\beta _3 }}{{\mathrm{{d}}t}}&= - \beta _3 (t)\sum \limits _k {g_k^2 } I_k (\Omega ,t) \nonumber \\&\quad - \beta _1 (t)\sum \limits _k {g_k^2 e^{i2kd} } I_k (\Omega ,t) \nonumber \\&\quad - \beta _2 (t)\sum \limits _k {g_k^{} g_k^{(0)} e^{ikd} e^{i(\Omega - \Omega _0 )t} } I_k (\Omega _0 ,t) , \end{aligned}$$

(A6)

where

$$\begin{aligned} \begin{aligned} I_k (\Omega ,t)&= \int \limits _0^t {e^{ - i(\omega _k - \Omega )(t - t')} \mathrm{{d}}t'} = \int \limits _0^t {e^{ - i(\omega _k - \Omega )\tau } d\tau } \\&\approx \int \limits _0^\infty {e^{ - i(\omega _k - \Omega )\tau } d\tau } = \pi \delta (\omega _k - \Omega ) - iP.v.\left( {\frac{1}{{\omega _k - \Omega }}} \right) \end{aligned} \end{aligned}$$

(A7)

where P.v. is a Cauchy principal value integral.

We can remove oscillating exponents in (A4)–(A6) with the aid of the substitution

$$\begin{aligned} \begin{aligned} \beta _1 (t) = e^{i(\Omega - \Omega _0 )t/2} \bar{\beta }_1 (t) \\ \beta _2 (t) = e^{ - i(\Omega - \Omega _0 )t/2} \bar{\beta }_2 (t) \\ \beta _3 (t) = e^{i(\Omega - \Omega _0 )t/2} \bar{\beta }_3 (t) . \end{aligned} \end{aligned}$$

(A8)

In addition, we assume that the coupling constants $g_k$ are even functions of k ($g_k=g_{-k}$). Then, from (A4)–(A6), we obtain

$$\begin{aligned} \frac{{\mathrm{{d}}\bar{\beta }_1 }}{{\mathrm{{d}}t}}&= - \bar{\beta }_1 (t)\sum \limits _k {g_k^2 } I_k (\Omega ,t) - i\frac{{\Omega - \Omega _0 }}{2}\bar{\beta }_1 (t) , \nonumber \\&\quad - \bar{\beta }_2 (t)2\sum \limits _{k> 0} {g_k^{} g_k^{(0)} \cos (kd)} I_k (\Omega _0 ,t) \nonumber \\&\quad - \bar{\beta }_3 (t)2\sum \limits _{k > 0} {g_k^2 \cos (2kd)} I_k (\Omega ,t) , \end{aligned}$$

(A9)

$$\begin{aligned} \frac{{\mathrm{{d}}\bar{\beta }_2 }}{{\mathrm{{d}}t}}&= - \bar{\beta }_2 (t)\sum \limits _k {\left( {g_k^{(0)} } \right) ^2 } I_k (\Omega _0 ,t) + i\frac{{\Omega - \Omega _0 }}{2}\bar{\beta }_2 (t) \nonumber \\&\quad - \bar{\beta }_1 (t)2\sum \limits _{k> 0} {g_k^{(0)} g_k^{} \cos (kd)} I_k (\Omega ,t) \nonumber \\&\quad - \bar{\beta }_3 (t)2\sum \limits _{k > 0} {g_k^{(0)} g_k^{} \cos (kd)} I_k (\Omega ,t) \end{aligned}$$

(A10)

$$\begin{aligned} \frac{{\mathrm{{d}}\bar{\beta }_3 }}{{\mathrm{{d}}t}}&= - \bar{\beta }_3 (t)\sum \limits _k {g_k^2 } I_k (\Omega ,t) - i\frac{{\Omega - \Omega _0 }}{2}\bar{\beta }_3 (t) \nonumber \\&\quad - \bar{\beta }_1 (t)2\sum \limits _{k> 0} {g_k^2 \cos (2kd)} I_k (\Omega ,t) \nonumber \\&\quad - \bar{\beta }_2 (t)2\sum \limits _{k > 0} {g_k^{} g_k^{(0)} \cos (kd)} I_k (\Omega _0 ,t) . \end{aligned}$$

(A11)

The next step is to relate the coupling constants $g_k$ to the qubit decay rate of spontaneous emission into waveguide mode. In accordance with Fermi golden rule, we define the qubit decay rates by the following expressions:

$$\begin{aligned}&\Gamma = 2\pi \sum \limits _k {g_k^2 \delta (\omega _k - \Omega )} \end{aligned}$$

(A12)

$$\begin{aligned}&\Gamma _0 = 2\pi \sum \limits _k {\left( {g_k^{(0)} } \right) ^2 \delta (\omega _k - \Omega _0 )}, \end{aligned}$$

(A13)

where

$$\begin{aligned} g_k^{} = \sqrt{\frac{{\omega _k D_{} ^2 }}{{2\hbar \varepsilon _0 V}}}; \end{aligned}$$

(A14)

D is the matrix element of the qubit’s dipole moment operator, and V is the effective volume where the interaction between qubit and electromagnetic field takes place. For 1D case, a summation over k is replaced by the integration over $\omega$ in accordance with the prescription

$$\begin{aligned} \sum \limits _k {} \Rightarrow \frac{L}{{2\pi }}\int \limits _{ - \infty }^\infty {dk} = \frac{L}{{2\pi }}2\int \limits _0^\infty {d\left| k \right| } = \frac{L}{{\pi \upsilon _g }}\int \limits _0^\infty {\mathrm{{d}}\omega _k }, \end{aligned}$$

(A15)

where L is a length of the waveguide, and we assumed a linear dispersion law, $\omega _k=v_gk$. The application of (A15) to, for example, (A12), allows the relation between a coupling constant $g_k$ and the decay rate $\Gamma$

$$\begin{aligned} g_\Omega ^{} = \sqrt{\frac{{\Omega D_{} ^2 }}{{2\hbar \varepsilon _0 V}}} = \left( {\frac{{v_g \Gamma }}{{2L}}} \right) ^{1/2}. \end{aligned}$$

(A16)

Therefore, we may relate the coupling constants $g_\Omega ^{(0)} ,g_{\Omega _2 }^{(0)} , g_{\Omega _0 }^{}$ with their respective decay rates

$$\begin{aligned}&g_\Omega ^{(0)} = \sqrt{\frac{{\Omega D_0 ^2 }}{{2\hbar \varepsilon _0 V}}} = \left( {\frac{\Omega }{{\Omega _0 }}} \right) ^{1/2} \left( {\frac{{v_g }}{{2L}}\Gamma _0 } \right) ^{1/2}, \qquad \end{aligned}$$

(A17)

$$\begin{aligned}&g_{\Omega _0 }^{} = \sqrt{\frac{{\Omega _0 D_{} ^2 }}{{2\hbar \varepsilon _0 V}}} = \left( {\frac{{\Omega _0 }}{\Omega }} \right) ^{1/2} \left( {\frac{{v_g }}{{2L}}\Gamma } \right) ^{1/2}, \qquad \end{aligned}$$

(A18)

$$\begin{aligned}&g_{\Omega _0 }^{(0)} = \sqrt{\frac{{\Omega _0 D_0 ^2 }}{{2\hbar \varepsilon _0 V}}} = \left( {\frac{{v_g }}{{2L}}\Gamma _0 } \right) ^{1/2}.\qquad \end{aligned}$$

(A19)

Now, we can calculate the different terms in (A9)–(A11). We begin with the sum in the first line in (A9)

$$\begin{aligned} \begin{aligned} \sum \limits _k {g_k^2 } I_k (\Omega ,t)&= \sum \limits _k {g_k^2 } \left( {\pi \delta (\omega _k - \Omega ) - iP.v.\left( {\frac{1}{{\omega _k - \Omega }}} \right) } \right) \\&= \frac{\Gamma }{2} - iP.v.\sum \limits _k \left( {\frac{{g_k^2 }}{{\omega _k - \Omega }}} \right) \approx \frac{\Gamma }{2} , \end{aligned} \end{aligned}$$

(A20)

where $\Gamma$ is given in (A12).

The second term in (A20) gives rise to the shift of the qubit frequency. Therefore, we incorporate it in the renormalized qubit frequency and will not write it explicitly any more. The sum in the second line in (A9) is calculated as follows:

$$\begin{aligned} \begin{aligned}&2\sum \limits _{k> 0} {g_k^{} g_k^{(0)} \cos (kd)} I_k (\Omega _0 ,t) \\&\quad = \frac{L}{{\upsilon _g }}\int \limits _0^\infty {g_k^{(0)} g_k^{} \cos (kd)\delta (\omega _k - \Omega _2 )\mathrm{{d}}\omega _k } \\&\qquad - 2i P.v.\sum \limits _{k > 0} \left( {\frac{{g_k^{} g_k^{(0)} \cos (kd)}}{{\omega _k - \Omega _{_0 } }}} \right) \\&\quad = \frac{L}{{\upsilon _g }}g_{_{\Omega 0} }^{(0)} g_{\Omega _0 }^{} \cos (k_0 d) - i\frac{L}{{v_g \pi }}g_{\Omega _0 }^{} g_{\Omega _0 }^{(0)} P.v.\int \limits _0^\infty {\mathrm{{d}}\omega } \frac{{\cos \left( {\frac{\omega }{{v_g }}d} \right) }}{{\omega - \Omega _0 }} . \end{aligned} \end{aligned}$$

(A21)

For principle value integral in (A21), we obtain with a good accuracy (see Appendix B)

$$\begin{aligned} P.v.\int \limits _0^\infty {\mathrm{{d}}\omega } \frac{{\cos \left( {\frac{\omega }{{v_g }}d} \right) }}{{\omega - \Omega }} \approx - \pi \sin \left( {\frac{\Omega }{{v_g }}d} \right) = - \pi \sin \left( {k_\Omega d} \right) . \end{aligned}$$

(A22)

As is shown in Appendix B, the result (A22) is obtained by a continuation of the frequency to negative axis, the trick which lies at the mathematical background of the Wigner–Weisskopf approximation.

Therefore, we finally obtain

$$\begin{aligned} \begin{aligned} 2\sum \limits _{k > 0} {g_k^{} g_k^{(0)} \cos (kd)} I_k (\Omega _0 ,t)&= \frac{L}{{\upsilon _g }}g_{_{\Omega _0 } }^{(0)} g_{\Omega _0 }^{} e^{ik_0 d} \\&= \frac{1}{2}\left( {\frac{{\Omega _0 }}{\Omega }} \right) ^{1/2} \sqrt{\Gamma \Gamma _0 } e^{ik_0 d} . \end{aligned} \end{aligned}$$

(A23)

Similar calculations give for the last sum in (A9)

$$\begin{aligned} 2\sum \limits _{k > 0} {g_k^2 \cos (2kd)} I_k (\Omega ,t) = \frac{\Gamma }{2}e^{2ikd}. \end{aligned}$$

(A24)

Collecting together (A20), (A23), and (A24), we write the final form of Eq. (A9)

$$\begin{aligned} \begin{aligned} \frac{{\mathrm{{d}}\bar{\beta }_1 }}{{\mathrm{{d}}t}}&= - \frac{{\Gamma _0 }}{2}\bar{\beta }_1 (t) - i\frac{{\Omega - \Omega _0 }}{2}\bar{\beta }_1 (t) \\&\quad -\bar{\beta }_2 (t) \frac{1}{2}\left( {\frac{\Omega _0 }{{\Omega }}} \right) ^{1/2} \sqrt{\Gamma \Gamma _0 } e^{ik_0d}-\frac{\Gamma }{2} \bar{\beta }_3 (t) e^{2ikd}. \end{aligned} \end{aligned}$$

(A25)

Similar calculations for Eqs. (A10) and (A11) yield the following result:

$$\begin{aligned} \frac{{\mathrm{{d}}\bar{\beta }_2 }}{{\mathrm{{d}}t}}&= - \frac{{\Gamma _0 }}{2}\bar{\beta }_2 (t) + i\frac{{\Omega - \Omega _0 }}{2}\bar{\beta }_2 (t) \nonumber \\&- \frac{1}{2}\left( {\frac{\Omega }{{\Omega _0 }}} \right) ^{1/2} \sqrt{\Gamma \Gamma _0 } e^{ikd} \left( {\bar{\beta }_1 (t) + \bar{\beta }_3 (t)} \right) \end{aligned}$$

(A26)

$$\begin{aligned} \frac{{\mathrm{{d}}\bar{\beta }_3 }}{{\mathrm{{d}}t}}&= - \frac{\Gamma }{2}\bar{\beta }_3 (t) - i\frac{{\Omega - \Omega _0 }}{2}\bar{\beta }_3 (t) - \bar{\beta }_1 (t)\frac{\Gamma }{2}e^{2ikd} \nonumber \\&- \bar{\beta }_2 (t)\frac{1}{2}\left( {\frac{{\Omega _0 }}{\Omega }} \right) ^{1/2} \sqrt{\Gamma \Gamma _0 } e^{ik_0 d} . \end{aligned}$$

(A27)

Appendix B: Proof of Eq. (A22)

The integral

$$\begin{aligned} \int \limits _0^\infty {\mathrm{{d}}\omega } \frac{{\cos \left( {\frac{\omega }{{v_g }}d} \right) }}{{\omega - \Omega }} = \int \limits _0^\infty {\mathrm{{d}}x} \frac{{\cos \left( {ax} \right) }}{{x - 1}}, \end{aligned}$$

where $a=k_\Omega d$, can be expressed in terms of sine and cosine integrals, ci and si [32]

$$\begin{aligned} \begin{aligned} \int \limits _0^\infty {\mathrm{{d}}x} \frac{{\cos \left( {ax} \right) }}{{x - 1}}&= - \cos (a)ci(a) - \sin (a)[si(a) + \pi ] \\&= - \cos (a)Ci(a) - \sin (a)\Big [Si(a) + \frac{\pi }{2}\Big ] , \end{aligned} \end{aligned}$$

(B1)

where

$$\begin{aligned} Ci(a) = - \int \limits _a^\infty {} \frac{{\cos t}}{t}\mathrm{{d}}t;\;Si(a) = \int \limits _0^a {} \frac{{\sin t}}{t}\mathrm{{d}}t. \end{aligned}$$

(B2)

The integrand in left-hand side in (B1) has a singular point at $x=1$ which manifests itself as a singularity of Ci(a) at $a\rightarrow 0$ in right-hand side in (B1). Therefore, we calculate integral (B1) as Cauchy principal value integral

$$\begin{aligned}&P.v. \int \limits _0^\infty {\frac{{\cos ax}}{{x - 1}}} \mathrm{{d}}x\approx P.v.\int \limits _{ - \infty }^\infty {\frac{{\cos ax}}{{x - 1}}} \mathrm{{d}}x\nonumber \\&\quad = P.v.\int \limits _{ - \infty }^\infty {\frac{{\cos a(t + 1)}}{t}} \mathrm{{d}}t \nonumber \\&\quad = \cos a\;P.v.\int \limits _{ - \infty }^\infty {\frac{{\cos at}}{t}} \mathrm{{d}}t - \sin a\;P.v.\int \limits _{ - \infty }^\infty {\frac{{\sin at}}{t}} \mathrm{{d}}t \end{aligned}$$

(B3)

$$\begin{aligned}&P.v.\int \limits _{ - \infty }^\infty {\frac{{\cos at}}{t}} \mathrm{{d}}t = \mathop {\lim }\limits _{R \rightarrow \infty } \mathop {\lim }\limits _{\varepsilon \rightarrow 0} \left( {\int \limits _{ - R}^{ - \varepsilon } {} \frac{{\cos at}}{t}\mathrm{{d}}t + \int \limits _\varepsilon ^R {} \frac{{\cos at}}{t}\mathrm{{d}}t} \right) \nonumber \\ \end{aligned}$$

(B4)

$$\begin{aligned}&P.v.\int \limits _{ - \infty }^\infty {\frac{{\sin at}}{t}} \mathrm{{d}}t = \mathop {\lim }\limits _{R \rightarrow \infty } \mathop {\lim }\limits _{\varepsilon \rightarrow 0} \left( {\int \limits _{ - R}^{ - \varepsilon } {} \frac{{\sin at}}{t}\mathrm{{d}}t + \int \limits _\varepsilon ^R {} \frac{{\sin at}}{t}\mathrm{{d}}t} \right) .\nonumber \\ \end{aligned}$$

(B5)

Fig. 13

Comparison of principle value $F_1$ (right-hand side of (B8) with the “exact” expression $F_2$ (right-hand side of (B1))

Since

$$\begin{aligned} \int \limits _{ - R}^{ - \varepsilon } {} \frac{{\cos a t}}{t}\mathrm{{d}}t = - \int \limits _\varepsilon ^R {} \frac{{\cos a t}}{t}\mathrm{{d}}t \end{aligned}$$

and

$$\begin{aligned} \int \limits _{ - R}^{ - \varepsilon } {} \frac{{\sin a t}}{t}\mathrm{{d}}t = \int \limits _\varepsilon ^R {} \frac{{\sin a t}}{t}\mathrm{{d}}t, \end{aligned}$$

we obtain

$$\begin{aligned}&P.v.\int \limits _{ - \infty }^\infty {\frac{{\cos a t}}{t}} \mathrm{{d}}t = 0 \end{aligned}$$

(B6)

$$\begin{aligned}&P.v.\int \limits _{ - \infty }^\infty {\frac{{\sin at}}{t}} \mathrm{{d}}t = \mathop {\lim }\limits _{R \rightarrow \infty } \mathop {\lim }\limits _{\varepsilon \rightarrow 0} \left( {2\int \limits _\varepsilon ^R {} \frac{{\sin at}}{t}\mathrm{{d}}t} \right) = \nonumber \\&\mathop {\lim }\limits _{R \rightarrow \infty } \left( {2\int \limits _0^R {} \frac{{\sin at}}{t}\mathrm{{d}}t} \right) = \mathop {\lim }\limits _{R \rightarrow \infty } \left( {2Si(aR)} \right) = \pi . \end{aligned}$$

(B7)

The final result in (B7) is due to the known relation [32]: $\mathop {\lim }\limits _{x \rightarrow \infty } Si(x) = \frac{\pi }{2}$.

Therefore, we finally obtain

$$\begin{aligned} P.v.\int \limits _0^\infty {\frac{{\cos ax}}{{x - 1}}} \mathrm{{d}}x \approx - \pi \sin a. \end{aligned}$$

(B8)

Below, in Fig. 13, we compare the kd-dependence of (B1) with that of (B8). We see a noticeable discrepancy for $kd<\pi /4$. For $kd>\pi /2$, two curves are almost identical.