All-optical pulse switching with a periodically driven dissipative quantum system

All-optical switching used to switch the input optical signals without any electro-optical conversion plays a vital role in the next generation of optical information processing devices. Even all-optical switchings (AOSs) with continuous input signals have been widely studied, all-optical pulse switchings (AOPSs) whose input signals are pulse sequences have rarely been investigated because of the time-dependent Hamiltonian, especially for dissipative quantum systems. In this paper, we propose an AOPS scheme, where a strong pulsed field is used to switch another pulsed input signal. With the help of Floquet-Lindblad theory, we identify the control field that can effectively turn on/off the input signal whose amplitude envelope is a square-wave (SW) pulse train in a three-level dissipative system. By comparing the properties of the AOPSs controlled by a continuous-wave (CW) field and an SW control field, we find that the SW field is more suitable to be a practical tool for controlling the input SW signal. It is interesting to impress that the switching efficacy is robust against pulse errors. The proposed protocol is readily implemented in atomic gases or superconducting circuits and corresponds to AOPSs or all-microwave pulse switchings.

All-optical switchings, in which the switching is produced by using a strong control field to control the propagation (or scattering) of an input probe field, have great potential for use in optical computation and communication [7]. Most existing theoretical studies are based on the continuous-wave (CW) model that assumes continuous monochromatic input fields [13,14], but practical applications and experimental demonstrations of switching usually employ pulses [15][16][17][18]. For a few specific single (not pulse train) pulses, some numerical simulations have been carried out [19][20][21][22]. However, to the best of our knowledge, the studies on AOPS are rare, especially the analytic solutions, because the time-dependent Floquet-Lindblad equation makes them a challenging task.
In this work, we present a theory of the AOPS for an input field whose amplitude envelope is an SW sequence in a three-level dissipative system. A variety of effects can be employed for switching [23][24][25][26]. The one we applied here is the transparency window phenomena [27,28]. The on (off) state is realized when the control field intensity is sufficiently strong (zero) to J 21 Fig. 1. A three-level system denoted by |0 , |1 and |2 with energy 0,1,2 couples with electromagnetic fields confined in a 1D transmission line. An input field with amplitude 0 couples states |0 and |1 with a detuning Δ = 1 − 0 − and the transmission coefficient is ′ .
is the amplitude of the scattering wave, propagating in both directions (forward and backward). The time dependence of the amplitude of the input field is approximated by an SW envelope with period and a duty cycle of 50%. A strong control field resonantly couples states |1 and |2 . 21 ( 10 ) is the population damping rate from state |2 (|1 ) to |1 (|0 ). 22 ( 11 is the dephasing rate of state |2 (|1 ). We consider a three-level atom interacting with a field of an electromagnetic 1D wave [33,34]. The amplitude of the input probe field is modulated by an SW envelope with period , as shown in Fig. 1. In the semiclassical theory of quantum optics, the field of the input wave is the frequency and k is the wave number) induces the atomic polarization. The atom placed at = 0 generates waves ( , ) = ( ) | |− , propagating in forward or backward direction. Here we assume the general scenario in which the atom size is negligibly small as compared with the wavelength [34] and investigate elastic Rayleigh scattering in which the incident and the scattered waves have the same frequency. It is convenient to define reflection and transmission coefficients and ′ according to ( ) = − 0 ( ) and 0 ( ) + ( ) = ′ 0 ( ), therefore, ′ + = 1. And is proportional to the off-diagonal element 10 = Tr[ 10 ] with transition operator 10 = |1 0| and density matrix [34,35], where is a constant determined by experimental device and Ω is the Rabi-frequency of the input field. For a traditional AOS [36,37], a strong CW control field is used to switch on/off the transmission of the input CW probe field. With the control off, the input is reflected from the transition line, and when the control is on, the input is transmitted through the transmission line and the reflection coefficient is almost zero (i.e., ≈ 0). Inspired by these, to switch on/off the transmission of the pulsed input field, we introduce another auxiliary level |2 and a strong control field with frequency = 2 − 1 . However, the characteristics of the control field (such as, continuous or pulsed field) are still unclear, and further research is needed. In the following, we will explore this problem in detail with both numerical and analytical methods by using Floquet-Lindblad theory. Here we take the ladder-type three-level system as an example. In fact, it can also be Λ-type or -type, which is common in atomic gases [38].

Analytical results of periodically driven dissipative quantum systems
In this section, we investigate the analytical results of periodically driven dissipative quantum systems under high-frequency expansions by using Floquet-Lindblad theory and Van Vleck perturbation theory [30].

Formula
Firstly, we introduce the general description for the nonequilibrium steady states (NESSs), which are the long-time dynamics of periodically driven dissipative quantum systems. For a quantum system defined on an N-dimensional Hilbert space, 0 denote the time-independent Hamiltonian with eigenenergies { } =1 and eigenstates {| } =1 . Here 1 < 2 < · · · < . The effect of the driving is represented by a time-dependent Hamiltonian ext ( ) with period : ext ( + ) = ext ( ). The total Hamiltonian is ( ) = 0 + ext ( ) and hence ( + ) = ( ). Thus the Fourier series of ( ) can be written as [39] with Floquet frequency = 2 / .
To study dissipative systems, we consider the density operator ( ), within the Born-Markov approximation [40], whose dynamics is described by the Floquet-Lindblad equation [31,41,42] (ℏ = 1 throughout this work) with { , } = + and projection operator ≡ | |. The long-time dynamic of Eq. (3) is NESS, which emerges in a balance of the energy injection by the periodic driving and the energy dissipation. The general description for the NESS under high Floquet frequency is given by [30] ( ) = G ( ) L eff −G (0) (0).
The time-dependent part G ( ) is the micromotion operator periodic in time G( + ) = G( ).
On the basis of the high-frequency expansion, i.e., the Floquet frequency is greater than the energy scales of 0 , the micromotion operator is given by The time-independent part L eff is represented by the effective Hamiltonian with Focusing on the leading-order contribution, Eq. (5) has a simple explicit formula [31] ness ( ) =˜ + MM ( ) in which both MM ( ) and FE are ( −1 ). MM ( ) and FE are the micromotion and Floquetengineering parts, respectively. The micromotion part MM ( ) is where˜ is characterized by There are two properties of MM ( ): (i) MM ( + ) = MM ( ) and it contributes to oscillations of physical observable, and (ii) MM ( ) does not contribute to the time averages of physical observable for one period of oscillation. The time-independent Floquet-engineering part FE is given by and | | = 0 for all , where Δ eff ≡ eff − 0 = ( −1 ),˜ = |˜ | and = ( + )/2. Floquet-engineering part FE describes how the effective Hamiltonian changes physical observable from their values in static counterpart and contributes to the time-averaged quantities. These formulas provide powerful tools for analyzing the NESSs of periodically driven dissipative quantum systems.

Dissipative two-level systems
We now calculate the NESS of a dissipative two-level system with the control field off. As shown in Fig. 1, a generic two-level system denoted by |0 , |1 driven by an SW-pulse input field with detuning Δ and time-dependent Rabi-frequency Ω ( + ) = Ω ( ). The Hamiltonian can be written as We have adopted the rotating wave approximation by assuming , Ω ≪ , which is also the parameter ranges of many experiments. For example, in superconducting circuit (atom) experiments, the frequency of near-resonant microwave (laser) is several GHz (THz), however, the Floquet frequency and the Rabi frequency of control and probe fields are about several or hundreds MHz [43]. Note that high-frequency approximation, i.e., > Δ, Ω , do not break this condition.
To calculate the Floquet engineering part in Eq. (12), we should get Δ eff and On the basis of high-frequency expansion (more precisely, > Δ, Ω ), we neglect higher order terms [i.e., higher than ( −1 )]. Therefore, focusing on the leading-order contribution, the NESS of 10 ( ) is given by in which Re( 10 ) = 0. In Fig. 2, we present the dynamics of the off-diagonal element Im( 10 ) with parameters = 0.01, Ω /(2 ) = 0.5, Δ = 0, 10 /(2 ) = 1 and 11 /(2 ) = 0.2. We find that it oscillates with time and finally reaches NESS after a sufficiently long time. Moreover, in the inset of Fig. 2, we compare the numerical and analytical results of Im( 10 ) and find that they agree well with each other. Indeed, the micromotion part of Im( 10 ) is a periodic triangular pulse predicted by Eq. (29).

Dissipative Floquet three-level systems
In this subsection, we investigate the other situation: the input is transmitted through the transmission line with a strong control field on and the reflection coefficient should be as small as possible. Based on the above two-level system, we further consider another level |2 with energy 2 > 1 > 0 , as shown in Fig. 1. Physically, there are dissipative rates and we set population damping rate 21 /(2 ) = 1.2 and dephasing rate 22 /(2 ) = 0.2. Other parameters are the same as in Sec. 3.2. Note that here 20 = 0, because the transition between levels |0 and |2 is forbidden. A control field with frequency couples levels |1 and |2 resonantly, i.e., 2 − 1 = . In this work, we consider two scenarios: (i) CW control field with constant Rabi-frequency Ω , and (ii) SW control field with time-dependent Rabi-frequency Ω ( ). For the CW control field with constant Rabi-frequency, as the lower inset shown in Fig. 3, the Hamiltonian of the system is with Ω ( ) is given by Eq. (14). We have adopted the rotating wave approximation by assuming , Ω ≪ , and Ω ≪ . For superconducting circuit (atom) systems, / 10 and / 10 are several thousands (millions). Similarly, the Floquet matrix blocks are Substituting Eq. (33) to Eq. (11) and with Δ = 0, we have with˜ 01 =˜ * 01 ,˜ 12 =˜ * 21 ,˜ 02 =˜ * 20 , Γ 2 = 21 /2 + 22 and Γ = Γ 1 + Γ 2 . As the atom is initially in the ground level |0 ,˜ 00 (0) = 1,˜ 11 (0) =˜ 22 (0) = 0 and˜ 21 (0) = 0. Combined with˜ 00 +˜ 11 +˜ 22 = 1 and weak probe field condition (Ω ≪ Ω ), the first-order steady state solutions to Eqs. (36)-(40) arẽ 10 After straightforward calculations, one obtains the micromotion part MM ( ) and then the off-diagonal term 10 MM ( ): which is similar to Eq. (29) and the higher order terms than ( −1 ) are omitted, except for the amplitude of the periodic triangular pulse. It is convenient to give the NESS of 10 ( ): In Fig. 3, we present the numerical and analytical results of the off-diagonal element Im( 10 ) for Ω /(2 ) = 50, which is larger than Ω /(2 ) = 0.5 to satisfy the valid condition of Eqs. (41)-(43) but smaller than Floquet frequency /(2 ) = 100 to satisfy the condition of the high-frequency expansions. From the figure, we find that the system reaches NESS after a long time, in which Im( ness 10 ) periodically oscillates in the form of a triangular wave with an average value close to zero, as predicted by Eq. (45). Moreover, we find from the figure that sometimes Im( ness 10 ) is far away from zero, because of the large amplitude of the triangular wave. So the reflection coefficient ( ∝ Im ness 10 ) of the input pulse controlled by the CW field is also time-dependent, and sometimes far away from zero, which is not a valid ON state for switching. Therefore, the CW control field isn't a practical switching for the pulsed input field. In the following, we alternatively study the properties of the switching with an SW control field.

Square-wave (SW) control field
Assuming the SW envelope of strong control field has the same frequency and initial phase with that of the input field, as the lower inset shown in Fig. 4. The Hamiltonian is where Similarly, the Floquet matrix blocks are Note that the time-dependent parts of H (i.e., ≠0 ) include large elements Ω , which are beyond the scope of the high-frequency expansion formula shown in Sec. 3.1. However, for weak control field (such as, Ω ≪ ), the formula is valid. Omitting the higher order terms than ( −1 ), the micromotion part MM ( ) of 10 is Note that here the results of˜ cannot be simply given by replacing Ω in Eqs. (41)- (43) with Ω /2, because Eqs. (41)- (43) is established under the condition that Ω ≫ Ω . However, the numerical results of˜ for any Ω can be obtained by calculating Eq. (11) with 0 in Eq. (50). In Fig. 4, we present the numerical and analytical results obtained from Eq. (53) for Ω /(2 ) = 10. From the figure, we find that the system reaches NESS after a long time, and with the SW control field on, the reflection coefficient of the input field is also time-dependent but the oscillation amplitude is very small, which indicates that the SW control field may be a practical switching for the pulsed input field. To explore this problem more comprehensively, in the following, we further study the properties of the switching controlled by a CW/SW field with a wide range of intensity.

All-optical pulse switching
In the above section, we have obtained the analytical expressions of the NESSs of the periodically driven dissipative systems under special conditions. These analytical results help us to better understand the NESS of the periodically driven dissipative system, which is the key issue in designing AOPS. For the dissipative two-level system with the control filed off, Eq. (31) gives the analytical results in a wide range of parameters. However, for the dissipative three-level systems with SW control field, the time-dependent parts of H (i.e., n≠0 ) include large elements Ω cn , which break down the condition of the formula in Sec. 3.1. In this section, to comprehensively investigate the properties of the AOPS in a wide range of Ω , we numerically integrate the   Switching has two states: (i) OFF state, where the input field is reflected by the system with control field off and (ii) ON state, where the input field transmits through the system with a The OFF/ON power ratiō off /¯ on of the switchings controlled by a CW field (blue line) and an SW field (red line) as a function of Ω . Obviously, within the parameter range shown in the figure, the¯ off /¯ on value of the switching controlled by the SW field is larger than that of the switching controlled by the CW field. strong control field on, i.e., the reflection coefficient → 0. For the OFF state, the reflection coefficient ∝ Im( ness 10 ) is given by Eq. (31). For the ON state, in Fig. 5(a) we compare the time evolutions of Im( 10 ) with a strong CW control field (blue line) on and a strong SW control field (red line) on. Both of them oscillate for a sufficiently long time and then reach NESSs. After reaching NESSs, the oscillation amplitude of Im( 10 ) controlled by the CW field is much larger than that controlled by SW field, which denotes that the reflection coefficient controlled by CW field oscillates violently and sometimes far away from zero. Moreover, in Fig. 5(b), we present the average and the standard deviation of Im( ness 10 ) controlled by a CW field (blue line) and an SW field (red line) as a function of Ω . We find that although their averages of Im( ness 10 ) are very close and approximate to 0, the standard deviations of Im( ness 10 ) are quite different and the latter is smaller. Therefore, the SW field is more suitable to switch the pulsed input field.
In Fig. 6, we present the results of time-domain numerical simulations of Im( 10 ) for a switching event corresponding to the one occurring in the AOPS controlled by a CW field [ Fig. 6(a)] and an SW field [ Fig. 6(b)]. One sees that the transients tend to NESSs and the oscillation amplitude of the ON state in Fig. 6(a) is larger than that in Fig. 6(b). Note that here the SW period is much larger than the rising/falling edge of the SW, and much smaller than the decoherence time of the system, i.e., = 0.01 ≪ 1/Γ ≈ 0.67 with total decay rate Γ = ( 10 + 21 )/2 + 11 + 22 = 1.5, resulting in at least 67 SW periods in a single switching time (i.e., the reaction time of the system to reach NESS). Therefore, when the SW control field is turned on/off within an SW period has a negligible effect on the switching effect. Additionally, the switching time is independent of whether the control field is CW or SW and depends on the decay rate of the system. Moreover, as a factor of merit for switching, the OFF/ON ratio between the reflected powers (R = | | 2 ) in the OFF state and that in the ON state R off /R on must be as large as possible. According to Eq. (1), R off /R on = |Im( ness 10 ) off | 2 /|Im( ness 10 ) on | 2 . Since the reflection coefficients in NESSs also oscillate with time, we calculate the average of reflected powersR off andR on . In Fig. 6(c), we further present theR off /R on ratio in the unit of dB as a function of Ω . It is clear that as the intensity of the control field increases, the values ofR off /R on controlled by the SW field increase, while the ones controlled by the CW field decreases. Moreover, the value ofR off /R on controlled by the SW field can be as high as nearly 70dB, which is much higher than that of the switching (about 50dB) in Ref. [44]. Obviously, for a high-performance switching, the strong SW control field is more advantageous. In the following section, we will explore the robustness ofR off /R on to the pulse errors. To estimate the robustness ofR off /R on to the pulse errors, we consider mismatch pulses between the control and the input fields, such as, assuming a nonoverlap time between the SW envelopes of the input and control fields, as the inset pulse sequences shown in Fig. 7. The black dashed line denotes that the opening time of the control field in one cycle completely covers the signal time of the input field and is redundant. On the contrary, the green dot-dashed line denotes that the time that the control field is turned on in one cycle does not completely cover the signal time of the input field. Remarkably, the OFF/ON powers ratiosR off /R on are quite close and show strong robustness against the mismatch pulses between control and the input fields, especially when the strength of the control field is in the range Ω ∈ [50, 100].

Conclusion
In conclusion, we propose an AOPS in a dissipative three-level system, in which an effective SW control field is used to switch an input field whose amplitude envelope is a periodic SW pulse. By comparing the NESSs of the dissipative systems driven by a CW and an SW field from analytical and numerical results, we find that the strong SW field with the same parameters as the input field (except for the Rabi-frequency), is a practical field for switching the input field and show strong robustness against the mismatch pulses between control and the input fields. Our theoretical protocol may be generalized to other pulse forms of the input field. This work may find potential applications in quantum computation processing and quantum networks based on all-optical devices.

Appendix: An all-optical pulse switching with electromagnetically induced transparency
Here, the all-optical pulse switching scheme is demonstrated with electromagnetically induced transparency parameters, i.e., same as the ones in the main text except for 21 /(2 ) = 0.1 and 22 /(2 ) = 0.01, which satisfy the condition that a transparent window is induced by destructive interference [45], as shown in Fig. 8 (a). Same as the main text, for an SW-pulse input field, in Fig. 8 (b) we compare the time evolutions of Im( 10 ) with a strong CW control field (blue line) on and a strong SW control field (red line) on. Clearly, after reaching NESSs, the oscillation amplitude of Im( 10 ) controlled by the CW field is much larger than that controlled by SW field. Moreover, in Fig. 8 (c), we present the average and the standard deviation of Im( ness 10 ) controlled by a CW field (blue line) and an SW field (red line) as a function of Ω . We find that the standard deviations of Im( ness 10 ) are quite different and the latter is much smaller. In Fig. 8  (d), we further present theR off /R on ratio in the unit of dB as a function of Ω . It is clear that the SW field is more suitable to switch the pulsed input field. Therefore, our all-optical pulsing scheme is also suitable to the electromagnetically induced transparency. The OFF/ON power ratio¯ off /¯ on of the switchings controlled by a CW field (blue line) and an SW field (red line) as a function of Ω with Δ = 0. Obviously, within the parameter range shown in the figure, the¯ off /¯ on value of the switching controlled by the SW field is larger than that of the switching controlled by the CW field. Note that the value of Ω starts from 11, because the intensity of the same electromagnetic field is halved under SW modulation. Other parameters are shown in Sec. 6.
Funding. This work is supported by the National Natural Science Foundation of China (Grant No.