Dynamical Symmetries and Symmetry-Protected Selection Rules in Periodically Driven Quantum Systems

Introduction. Over the last few decades, the light-matter interaction strength has been pushed to the ultrastrong coupling regime in optomechanical systems [1], quantum dots, atoms and molecules in optical or plasmonic cavities [2, 3, 4, 5, 6], and superconducting quantum circuits [7, 8, 9]. As standard nonlinear perturbation theory [10] becomes unfeasible under these conditions, Floquet response theory has been developed recently, describing systems that are subject to a strong but time-periodic driving field (of frequency $\Omega$), and a weak but arbitrary probe field [11, 12, 13, 14]. For a monochromatic probe of frequency $\omega_{p}$, system observables generate response frequencies $\omega_{p}+n\Omega$ termed Floquet bands [15].

Spatial symmetries give rise to appealing physical properties. Inversion symmetry results in selection rules for dipole transitions; particle-hole, chiral, and time-reversal symmetries establish the so-called periodic table, a classification scheme for topological insulators [16, 17, 18], and symmetries have an essential impact on transport properties [19, 20, 21, 22, 23, 24, 25]. For periodically driven systems, these spatial symmetries can be generalized to dynamical symmetries that can give rise to a generalized periodic table for topological insulators [26, 27] and new control mechanisms  [28, 29, 30, 31, 32, 33, 34, 35]. Dynamical symmetries have been used to control the coherent destruction of tunneling effect [36] and induce selection rules for high harmonic generation [37, 38, 39, 40].

In this Letter, we introduce a unified conceptual framework of selection rules based on general dynamical symmetries of periodically driven quantum systems as described by Floquet response theory [41]. Physically, the synchronization of symmetric quantum systems with the periodic driving gives rise to destructive interference effects in Floquet space and thus to forbidden transitions between Floquet states. This set of forbidden transitions defines the symmetry-protected selection rules that are robust against symmetry-preserving parameter variations. Specifically, there are four types of forbidden transitions ordered in increasing degree of complexity: (i) accidental dark states (aDSs), appearing for a specific combination of system parameters; (ii) symmetry-protected dark states (spDSs), which refer to the symmetry-protected absence of a complete transition line similar to symmetry-protected excitations of topological band structures; (iii) symmetry-protected dark bands (spDBs), which refer to the absence of a complete Floquet band due to a combination of spDSs; (iv) symmetry-induced transparency (siT), which refers to the vanishing transition intensity at the degeneracy of quasienergies. Except for the aDS, which is not symmetry related, we establish symmetry-protected selection rules for important dynamical symmetries, which are classified in Table 1.

Floquet response theory. We apply a semiclassical approach based on the general Hamiltonian

where $\hat{H}_{0}(t)=\hat{H}_{0}(t+\tau)$ describes a system driven by a periodic classical electromagnetic field of frequency $\Omega=2\pi/\tau$. The probe field is given by a continuum of photonic operators $a_{\omega}^{\dagger}$ with frequencies $\omega$, which are coupled via the dipole transition operator $\hat{V}$ with strength $\lambda$ to the driven system. The physical properties of $\hat{H}_{0}(t)$ are determined by the Floquet equation

where $\left|u_{\mu}(t)\right>=\left|u_{\mu}(t+\tau)\right>$ and $\epsilon_{\mu}$ are the corresponding Floquet states and quasienergies that generalize the concept of eigenstates and eigenenergies of time-independent systems. It is implicitly assumed that the driven system is weakly dissipative such that, for long times, it approaches the stationary state

which is diagonal in the Floquet basis and thus synchronizes with the driving $\rho(t)=\rho(t+\tau)$. Equation (3) is consistent with the Floquet-Redfield equation [42, 43, 44, 45] describing periodically driven open quantum systems. In the model calculations, we assume the special distribution $p_{\mu}\propto e^{-\beta\epsilon_{\mu}}$, i.e., a Floquet-Gibbs distribution, but all our predictions hold even if the Floquet-Gibbs distribution breaks down [45]. Strongly dissipative systems could be addressed by generalizing our approach to non-Hermitian Hamiltonians [18] or via the polaron transformation [46].

The interaction of $\hat{H}_{0}(t)$ with the probe field is treated using the input-output formalism and a perturbation expansion for small $\lambda$. The input field consists of a bichromatic probe field (of frequencies $\omega_{p,1}$ and $\omega_{p,2}=\omega_{p,1}+n\Omega$, integer $n$). As shown separately [14], the intensity change of the output field at frequency $\omega_{p,2}$ proportional to the coherence $\left<\hat{a}_{\omega_{p,2}}^{\dagger}\hat{a}_{\omega_{p,1}}\right>$ is given by $\Delta I_{\rm coh}(\omega_{p,2})=-i\tilde{\chi}_{n}(\omega_{p,1})\left<\hat{a}_{\omega_{p,2}}^{\dagger}\hat{a}_{\omega_{p,1}}\right>+\text{c.c.}$, where the susceptibility $\tilde{\chi}_{n}(\omega_{p,1})$ can be evaluated using Floquet response theory and reads

The index $n$ denotes the Floquet band, which describes nonelastic scattering of the probe field, and the dynamical dipole matrix elements read

The parameters $\gamma_{\nu,\mu}^{(m)}$ have been added phenomenologically and denote dephasing rates.

Unified conceptual framework of dynamical-symmetry-protected selection rules. We consider the following class of symmetry operations [26]:

where $\hat{\Sigma}$ is a time-independent spatial operator. By specifying $\hat{\Sigma}$, $t_{S}$, and $(\alpha_{S},\beta_{s}=\pm 1)$, one can define a set of dynamical symmetries. Applying Eq. (6) to the Floquet equation Eq. (2), one can identify relations between Floquet states $\mu$ and $\mu^{\prime}$:

These relations can be used to evaluate the dynamical dipole elements in Eq. (5). Imposing an invariance condition for the transition dipole operator $\hat{\Sigma}^{\dagger}\hat{V}\hat{\Sigma}=\alpha_{V}^{(S)}\hat{V}$ and using Eq. (7), we investigate symmetry-protected selection rules for rotational, parity, particle-hole, chiral and time-reversal symmetries.

Within the unified framework, we can establish symmetry-protected selection rules that are robust against symmetry-conserved variations and unique for strong light-matter interactions. Among others, we investigate dark states, which are defined by the condition $V_{\nu,\mu}^{(m)}=0$, such that the corresponding resonances in Eq. (4) vanish. This condition not only generalizes the dark state condition in the standard response theory to the strong-coupling regime for $n=0$ but also introduces distinct dark states effects for $n\neq 0$. All selection rules are a consequence of destructive interference due to the synchronization of the system state with the periodic driving: (i) The dark state condition can be fulfilled by special combinations of parameters, which we denote as an aDS, or (ii) as a consequence of a symmetry, which we denote as a spDS. (iii) An entire Floquet band can vanish because $\tilde{\chi}_{n}(\omega_{p})=0$ for specific $n$, which we denote as a spDB. (iv) By analyzing the susceptibility in terms of Eq. (7), we establish the condition for the siT, which is due to a destructive interference of two transitions with $V_{\nu,\mu}^{(m)}\neq 0$.

Rotational symmetry. With $\alpha_{S}=\beta_{S}=1$, a unitary $\hat{\Sigma}=\hat{R}$, and $t_{S}=t_{R}=\frac{\tau}{N}$ with a positive integer $N$, Eq. (6) defines a dynamical rotational symmetry [47] that gives rise to the eigenvalue equation $\left|u_{\mu}(t)\right>=\pi_{\mu}^{(R)}\hat{R}\;\left|u_{\mu}(t+t_{R})\right>$ with eigenvalues $\pi_{\mu}^{(R)}=e^{i2\pi m_{\mu}/N}$ and integer $m_{\mu}=\left\{0,N-1\right\}$. As shown in detail in the Supplemental Material [48], for a dipole transition operator with $\hat{R}^{\dagger}\hat{V}\hat{R}=\alpha_{V}^{(R)}\hat{V}$ and $\alpha_{V}^{(R)}=\pm 1$, the dynamical rotational symmetry establishes a sufficient condition for spDSs:

Applying Eq. (8) to evaluate the susceptibility in Eq. (4), we find

which is the condition for the complete disappearance of Floquet band $n$, i.e., a spDB. Physically, this effect appears as the stationary state Eq. (3) synchronizes with the driving field such that the density matrix adopts the dynamical rotational symmetry, i.e., $\rho(t+n/N\tau)=\hat{R}^{n}\rho(t)\hat{R}^{\dagger n}$.

As an example, we consider a benzene ring driven by circularly polarized light sketched in Fig. 1(a), which is described by a tight-binding Hamiltonian:

where $\left|e_{j}\right>$ denotes the excitation on site $j$ (defined modulo $6$), $J_{j,j}=E_{0}$ is the on-site energy, $J_{j,j^{\prime}}=\delta_{j,j^{\prime}\pm 1}J_{0}$ is the tunneling constant, and $f_{j}(t)=f_{\Omega}\cos(\Omega t+2\pi j/6)$ is the time-dependent tunneling strength with the driving amplitude $f_{\Omega}$. The driving terms are motivated by the Peierls substitution describing a vectorial current-gauge-field coupling $\boldsymbol{j}\cdot\boldsymbol{A}(t)$ [49] with a circularly rotating vector potential $\boldsymbol{A}(t)$. The dipole transition operator $\hat{V}=\sum_{j=1}^{N}d_{0}\left|e_{j}\right>\left<g\right|$ excites the ground state $\left|g\right>$ to the single-excitation manifold, whose quasienergies are depicted in Fig. 1(b). The stationary state is $\rho_{s}(t)=\left|g\right>\left<g\right|$ in agreement with Eq. (3), i.e., a Floquet-Gibbs state for low temperatures. A rotational symmetry is fulfilled for $N=6$ and $\hat{R}=\sum_{j=1}^{n}\left|e_{j+1}\right>\left<e_{j}\right|$.

In Fig. 1(c), we depict the susceptibility $\tilde{\chi}_{0}(\omega_{p})$ of the benzene model. The resonances of the dark states defined by Eq. (8) are marked by dashed lines (optically invisible), and only two transitions, $\hat{V}_{0,1}^{(0)}\hat{V}_{1,0}^{(0)}$ and $\hat{V}_{3,0}^{(1)}\hat{V}_{0,3}^{(-1)}$, are visible. An aDS can be found for $\hat{V}_{0,1}^{(0)}\hat{V}_{1,0}^{(0)}$ at $f_{\Omega}=1.5\Omega$. As a consequence of the spDB in Eq. (9), only Floquet bands $\tilde{\chi}_{n}(\omega_{p})$ with $n\mod 6=0$ appear.

Parity symmetry. A dynamical parity symmetry is a specification of the dynamical rotational symmetry with $N=2$ and a Hermitian operator $R^{\dagger}=R$ such that the spDS condition Eq. (8) and the spDB condition Eq. (9) are equally valid. The spDSs will be illustrated for the two-level system (TLS) in Eq. (13) along with the siT discussed below.

Particle-hole symmetry. A particle-hole symmetry is defined for $-\alpha_{S}=\beta_{S}=1$, $t_{S}=t_{P}=\tau N_{1}/2N_{2}$ with integers $N_{1}\in\left\{0,1\right\}$, $N_{2}\geq 1$, and $\hat{\Sigma}=\hat{P}\hat{\kappa}$ with a unitary operator $\hat{P}$ and the complex conjugation operator $\hat{\kappa}$, such that $\hat{P}\hat{H}^{*}(t+t_{P})\hat{P}=-\hat{H}(t).$ The particle-hole symmetry establishes a symmetry between the excitation and deexcitation processes and has its origin in fermionic systems, where adding and removing quasiparticles results in physically equivalent behaviors. Here we use the particle-hole symmetry in a general context. Using the particle-hole symmetry in Eq. (2), we find that each Floquet state $\left|u_{\mu}(t)\right>$ with quasienergy $\epsilon_{\mu}$ has its symmetry related partner $\left|u_{\mu^{\prime}}(t)\right>=\pi^{(P)}_{\mu}\hat{P}\left|u_{\mu}(t+t_{P})\right>^{*}$ with energy $\epsilon_{\mu^{\prime}}=-\epsilon_{\mu}$ and a gauge-dependent $\pi^{(P)}_{\mu}$. For $t_{P}=\tau/(2N_{2})$, the particle-hole symmetry gives rise to a rotational symmetry defined by $\hat{R}=\hat{P}\hat{P}$ and $t_{R}=\tau/N_{2}$ such that the dark state selection rules of the rotational symmetry apply. The particle-hole symmetry can give rise to a distinct dark state condition. For a dipole transition operator with $\hat{P}^{\dagger}\hat{V}^{*}\hat{P}=\alpha_{V}^{(P)}\hat{V}$, $\alpha_{V}^{(P)}=\pm 1$, $t_{P}=0,\tau/2$, and $\hat{P}^{*}\hat{P}=1$, the particle-hole symmetry results in $V_{\mu,\mu^{\prime}}^{(m)}=\alpha_{V}^{(P)}e^{im\Omega t_{P}}V_{\mu,\mu^{\prime}}^{(m)}$ for the symmetry-related states $\mu,\mu^{\prime}$, so that

as shown in detail in the Supplementary Material [48]. In contrast to Eq. (8), where each transition can vanish for an appropriate $m$, only transitions between symmetry-related states are affected by Eq. (10).

To illustrate Eq. (10), we use the dimer model sketched in Fig. 2(a), with the Hamiltonian given by

where $\hat{A}_{\alpha,\beta}\equiv\left|\alpha\right>\left<\beta\right|+\text{H.c.}$, and $g,e_{1},e_{2}$ and $f$ label the ground state, two single-excitation states, and the double-excitation state, respectively. $\Delta$ is the excitation gap, $J_{0}$ is the tunneling constant, and $h_{1}(t)=f_{\Omega}\cos(\Omega t)$ is the driving field. The $r$ term enhances higher-order dipole elements $V^{m\neq 0}_{\mu,\mu^{\prime}}$. The particle-hole symmetry is defined by $\hat{P}=\hat{A}_{g,f}+\hat{A}_{e_{1},e_{1}}-\hat{A}_{e_{2},e_{2}}$ and $t_{P}=0$. The quasienergy spectrum in Fig. 2(b) is symmetric with respect to $E=0$. The dipole transition operator is $\hat{V}=\hat{A}_{e_{1},f}+\hat{A}_{g,e_{1}}$, such that $\hat{P}^{\dagger}\hat{V}^{*}\hat{P}=-\hat{V}$. In Fig. 2(c), we depict the susceptibility in Eq. (4). According to the above considerations, the transitions between the particle-hole symmetry-related pairs vanish, i.e., $V_{1,4}^{(m)}=V_{4,1}^{(m)}=V_{2,3}^{(m)}=V_{3,2}^{(m)}=0$ for all $m$. These resonances are marked by dashed lines. The other transitions not affected by the symmetry constrain remain visible in Fig. 2(c).

Symmetry-induced transparency. The particle-hole symmetry can also give rise to a siT at the quasienergy crossing $\epsilon_{\mu}=\epsilon_{\mu^{\prime}}=0$ of the symmetry-related Floquet states $\mu,\mu^{\prime}$. While a spDS is generated by a vanishing dipole element, $V_{\lambda,\mu}^{(n)}=0$, the siT is generated by a destructive interference of two transitions with $V_{\lambda,\mu}^{(n)}\neq 0$. As shown in the Supplementary Material [48] in detail, for two distinct particle-hole symmetries $\hat{P}_{1}\neq\pm\hat{P}_{2}$, $\hat{P}_{i}^{2}=\mathbbm{1}$ and $\left[\hat{P}_{1},\hat{P}_{2}\right]=0$, the siT condition reads

where $t_{P_{i}}$ denote the reference times related to $\hat{P}_{i}$.

For illustration, we consider an ac-driven TLS sketched in Fig. 3(a) and described by the Hamiltonian

where $\hat{\sigma}_{x},\hat{\sigma}_{\rm z}$ are the Pauli matrices, $h_{x}$ is the tunneling amplitude, and $f_{\Omega}$ the driving strength. The TLS is weakly dissipative, as in the spin-boson model, such that it reaches the stationary state in Eq. (3). The dipole transition operator in Eq. (1) is $\hat{V}=\hat{\sigma}_{x}$. For $\hat{R}=\hat{\sigma}_{\rm x}$ and $t_{R}=\tau/2$, the TLS exhibits a dynamical parity symmetry defined above, which gives rise to the coherent destruction of tunneling effect at an exact quasienergy crossing, depicted in Fig. 3(b) at $f_{\Omega}\approx 2.4\Omega$ [36, 50], and enables the siT in the current context. Additionally, the TLS exhibits spDSs and spDBs according to Eq. (8) and Eq. (9) as $V_{\mu,\nu^{\prime}}^{(m)}=0$ for even $m$ because of the dynamical parity symmetry.

For the TLS, a particle-hole symmetry is defined for $\hat{P}_{1}=\hat{\sigma}_{\rm z}$ and $t_{P_{1}}=\tau/2$. For $h_{\rm x}=0$, a second particle-hole symmetry is given for $\hat{P}_{2}=\mathbbm{1}$ and $t_{P_{2}}=\tau/2$. As in this case $\epsilon_{\mu}=0$ and $\hat{P}_{i}\hat{\sigma}_{x}^{*}\hat{P}_{i}=(-1)^{i}\hat{\sigma}_{x}$, siT with $\tilde{\chi}_{n}(m\Omega)=0$ appears according to Eq. (12), and the response $\tilde{\chi}_{n}(\omega_{\rm p})$ is complete suppressed for all $n$. In Fig. 3(c), we consider $\tilde{\chi}_{0}(\omega_{p})$ for a finite but small $h_{\rm x}\ll\Omega$ such that the quasienergy degeneracy is lifted except of the crossing, and the particle-hole symmetry $\hat{P}_{2}$ is slightly broken. As a consequence, the siT is not complete but scales as $\tilde{\chi}_{n}(m\Omega)\propto h_{x}/\Omega$ at the crossing.

Time-reversal and chiral symmetries. A time-reversal symmetry (chiral symmetry) is defined by Eq. (6) for $\alpha_{S}=-\beta_{S}=1$ ($\alpha_{S}=\beta_{S}=-1$), arbitrary $t_{S}$, and $\hat{\Sigma}=\hat{T}\hat{\kappa}$, ($\hat{\Sigma}=\hat{C}$), where $\hat{T}$ ($\hat{C}$) is a unitary operator. As shown in the Supplementary Material [48], neither time-reversal symmetry nor chiral symmetry alone implies spDSs. However, the combination of time-reversal symmetry and chiral symmetry defines a particle-hole symmetry with $\hat{P}=\hat{C}\hat{T}$, and $t_{P}=t_{T}-t_{C}$. When they further fulfill $t_{T}-t_{C}\in\left\{0,\tau/2\right\}$, $\hat{C}^{*}\hat{C}=\mathbbm{1}$, $\hat{T}^{*}\hat{T}=\mathbbm{1}$, and $\left[\hat{C},\hat{T}\right]=0$, such that $\hat{P}^{*}\hat{P}=1$, spDSs appear because of the particle-hole symmetry. In general, the presence of any two symmetries out of particle-hole symmetry, chiral symmetry, and time-reversal symmetry implies the existence of the third one.

Conclusions. Using a unified conceptional framework based on Floquet response theory, we have predicted selection rules in periodically driven quantum systems, namely accidental dark states, symmetry-protected dark states, symmetry-protected dark bands, and symmetry-induced transparency. The latter three effects are protected by symmetries such that variations of symmetry preserving parameters do not destroy them. These symmetry-induced selection rules result from the destructive interference of a driven system synchronized to the periodic driving. The different effects have been illustrated in three example systems fulfilling different symmetries, demonstrating the flexibility and generality of our unified framework. The predicted selection rules are valid even for more complicated and realistic systems as long as the corresponding dynamical symmetries are fulfilled.

Our theoretical results are experimentally observable in systems that can reach the strong light-matter coupling regime such as cold-atom experiments [2] and superconducting circuits [51, 52, 53, 54]. For experiments with molecules, strong driving fields are necessary to generate high-order Floquet bands, but in cavity QED or plasmonic fields, the strong driving interaction condition can be relaxed for molecule ensembles interacting collectively with the light field [55, 56].

Acknowledgements. G. E. gratefully acknowledges financial support from the China Postdoc Science Foundation (Grant No. 2018M640054 ) and the Natural Science Foundation of China (Grant Nos. 11950410510 and U1930402), J. C. acknowledges support from the NSF (Grant Nos. CHE 1800301 and CHE1836913). The authors thank Tao Wang for helpful discussions.

Supplementary Material