Two-color electromagnetically induced transparency via modulated coupling between a mechanical resonator and a qubit

Two-color electromagnetically induced transparency via modulated coupling between a mechanical resonator and a qubit

Xin Wang Institute of Quantum Optics and Quantum Information, School of Science, Xi’an Jiaotong University, Xi’an 710049, China RIKEN, Wako-shi, Saitama 351-0198, Japan    Adam Miranowicz RIKEN, Wako-shi, Saitama 351-0198, Japan Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland    Hong-Rong Li Institute of Quantum Optics and Quantum Information, School of Science, Xi’an Jiaotong University, Xi’an 710049, China    Fu-Li Li Institute of Quantum Optics and Quantum Information, School of Science, Xi’an Jiaotong University, Xi’an 710049, China    Franco Nori RIKEN, Wako-shi, Saitama 351-0198, Japan Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA Institute of Quantum Optics and Quantum Information, School of Science, Xi’an Jiaotong University, Xi’an 710049, China RIKEN, Wako-shi, Saitama 351-0198, Japan RIKEN, Wako-shi, Saitama 351-0198, Japan Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland Institute of Quantum Optics and Quantum Information, School of Science, Xi’an Jiaotong University, Xi’an 710049, China RIKEN, Wako-shi, Saitama 351-0198, Japan Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA Institute of Quantum Optics and Quantum Information, School of Science, Xi’an Jiaotong University, Xi’an 710049, China RIKEN, Wako-shi, Saitama 351-0198, Japan RIKEN, Wako-shi, Saitama 351-0198, Japan Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland Institute of Quantum Optics and Quantum Information, School of Science, Xi’an Jiaotong University, Xi’an 710049, China RIKEN, Wako-shi, Saitama 351-0198, Japan Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA
July 12, 2019
Abstract

We discuss level splitting and sideband transitions induced by a modulated coupling between a superconducting quantum circuit and a nanomechanical resonator. First, we show how to achieve an unconventional time-dependent longitudinal coupling between a flux (transmon) qubit and the resonator. Considering a sinusoidal modulation of the coupling strength, we find that a first-order sideband transition can be split into two. Moreover, under the driving of a red-detuned field, we discuss the optical response of the qubit for a resonant probe field. We show that level splitting induced by modulating this longitudinal coupling can enable two-color electromagnetically induced transparency (EIT), in addition to single-color EIT. In contrast to standard predictions of two-color EIT in atomic systems, we apply here only a single drive (control) field. The monochromatic modulation of the coupling strength is equivalent to employing two eigenfrequency-tunable mechanical resonators. Both drive-probe detuning for single-color EIT and the distance between transparent windows for two-color EIT, can be adjusted by tuning the modulation frequency of the coupling.

pacs:
42.50.Ar, 42.50.Pq, 85.25.-j

I Introduction

Superconducting quantum circuits (SQCs) Makhlin et al. (2001); Devoret and Martinis (2004); You and Nori (2005); Clarke and Wilhelm (2008); You and Nori (2011); Buluta et al. (2011); Xiang et al. (2013); Schoelkopf and Girvin (2008) are ideal artificial platforms for studying microwave photonics Gu et al. (2017). Many quantum-optical effects, such as quantum Rabi oscillations Johansson et al. (2006); Garziano et al. (2015), electromagnetically induced transparency (EIT) Murali et al. (2004); Dutton et al. (2006); Ian et al. (2010); Chang et al. (2011); Jing et al. (2015); Gu et al. (2016), Autler-Townes splitting Sillanpää et al. (2009); Li et al. (2012); Novikov et al. (2013); Suri et al. (2013), and photon blockade Hoffman et al. (2011); Lang et al. (2011), have been successfully demonstrated with SQCs.

In contrast to natural atoms and optical cavities, the basic elements (i.e., multi-level superconducting systems and resonators) in SQCs can freely be designed and controlled for various purposes in microwave photonics Gu et al. (2017) and quantum-information technologies Wendin (2017).

Most commonly, the coupling between a superconducting qubit and a single-mode microwave resonator field is transverse You and Nori (2003); Gu et al. (2017), where the dipole moment of the qubit interacts with the electric (or magnetic) field of the resonator mode, and therefore being an exact analog of the standard quantum Rabi model in cavity quantum electrodynamics (QED) experiments Scully and Zubairy (1997). The quantum dynamics of such systems has been extensively studied for decades due to its potential applications in, e.g., quantum information processing and quantum optics Liu et al. (2006, 2004); Fink et al. (2008); Hofheinz et al. (2008); Wang et al. (2008).

In recent years, some theoretical and experimental research studies have been devoted to SQCs with another type of interaction form, i.e., the so-called longitudinal coupling Kerman (2013); Liu et al. (2014a); Zhao et al. (2015); Richer and DiVincenzo (2016); Richer et al. (2017). In a circuit-QED system with longitudinal interaction, the qubit-transition frequency is modulated by a quantized field, and the Pauli qubit operator couples with a quadrature field operator Richer et al. (2017). Compared with the transverse coupling Stassi and Nori (2018), the longitudinal coupling has its inherent advantages since the interaction term commutes with the qubit operator . For example, there is no Purcell decay and residual interactions between a qubit and its resonator. By using SQCs with longitudinal-coupling, one can realize various quantum-control tasks, such as error-correction codes Fowler et al. (2012); Stassi et al. (2017) or multiexcitation generation Garziano et al. (2015); Wang et al. (2017a); Stassi et al. (2017); Kockum et al. (2017) among many other applications Gu et al. (2017).

Recently, several studies have been focused on systems with parametrically-modulated longitudinal coupling, where the interaction strength was rather not constant but modulated in time at certain frequencies. This modulated interaction can be viewed as a qubit-state-dependent drive on a resonator. If the modulation rate is equal to the resonator frequency, qubit states can be readout rapidly via quantum nondemolition (QND) measurements Didier et al. (2015). Moreover, it is possible to obtain a high-fidelity controlled-phase gate, if modulating the longitudinal coupling between two remote qubits and a common resonator Royer et al. (2017). All these studies indicate that the modulated longitudinal coupling has its own advantages over the transverse and constant longitudinal interactions, and provides another way to achieve better quantum control and engineering. However, as discussed in Refs. Didier et al. (2015); Liao and Tian (2016); Cirio et al. (2017), it is not easy to obtain such modulated couplings in either natural or artificial systems.

In this paper, we describe possible sideband transitions and the optical response in a system with a parametrically-modulated longitudinal coupling which, to our knowledge, has not been discussed in previous studies. We start our discussions by proposing two possible circuit layouts, where superconducting qubits are longitudinally coupled to nanomechanical resonators (NAMRs) via an external flux Shevchuk et al. (2017). The coupling can be conveniently modulated in time by changing external magnetic fields. Considering a transverse driving field of a qubit, we find that a sideband transition can be split into several asymmetric parts if the modulation is sinusoidal.

Assuming that a resonant probe field is also applied to a qubit, we demonstrate that both single- and two-color (bichromatic) EIT Wang et al. (2003, 2014); Yan et al. (2013) can be observed. For single-color EIT, parametric modulation is equal to a flexible NAMR with a tunable eigenfrequency and, therefore, the drive-probe detuning of the EIT dip can be conveniently tuned. For two-color EIT, two transparent windows result from the parametric modulation of the longitudinal coupling, and their distance of two transparent windows can be adjusted by changing the modulation frequency, rather than sweeping two control frequencies in a conventional bichromatic EIT system Wang et al. (2006); Moiseev and Ham (2006); Yan et al. (2013).

There are various potential applications of EIT based on circuit-QED systems, such as optical switching Bajcsy et al. (2009); Xia and Twamley (2013); Gu et al. (2017), controlling slow light for information storage, demonstrating single-photon router devices Hoi et al. (2011); Leung and Sanders (2012), and controlling photon transmission through a circuit-QED system He et al. (2007); Liu et al. (2014b). Two-color EIT can be employed for entangling photons via cross-phase modulation and slowing photons at different frequencies Lukin and Imamoğlu (2000); Wang et al. (2006); Li et al. (2008). Our results can be helpful to study the dynamics for systems with time-dependent longitudinal coupling, and applications based on EIT in microwave photonics Gu et al. (2017).

The outline of the paper is as follows. In Sec. II, we describe a possible approach to mediate a flux (transmon) qubit with a NAMR via a modulated longitudinal coupling. In Sec. III, we derive an analytical Hamiltonian describing sideband-transition splitting. In Sec. IV, we discuss single- and two-color EIT, and show how to tune these two effects by changing drive-field parameters. Our final discussions and conclusions are presented in Sec. V.

Ii Model

A possible circuit-QED implementation of the time-dependent longitudinal interaction has been discussed in Ref. Didier et al. (2015). Specifically, a transmon qubit was assumed to interact with a transmission-line resonator via a Josephson junction inserted at the end of a central conductor. By modulating an external flux through the superconducting quantum interference device (SQUID) loop of the transmon qubit, a desired sinusoidally-modulated coupling can be obtained. However, the qubit transition frequency is also perturbed by a time-dependent control flux (with a frequency of about tens of MHz), which should be avoided in certain cases.

Here we demonstrate another possible hybrid circuit layout to realize such modulated interactions between high-frequency NAMR and superconducting qubits.

ii.1 Flux-mediated coupling between SQUID and NAMR

Figure 1: Two Josephson junctions with an identical capacitance form a SQUID loop in the - plane. A NAMR (represented by the red curve), with length and mass , is embedded in the loop and vibrates in the -direction with amplitude . The static magnetic field perpendicular to the - plane produces a static flux for the SQUID. Additionally, there is a time-dependent magnetic field [parallel to the - plane] applied to the loop area produced by the NAMR. The total flux through the SQUID is modulated by the vibrational motions of NAMR.

We start our discussions by describing the interaction between a SQUID and a NAMR. As shown in Fig. 1, two Josephson junctions, each shunted by a capacitance , are connected by a loop in the - plane. The total charging energy for two junctions is . The NAMR with length is coated with a superconductor and fabricated into the loop Etaki et al. (2008); Poot et al. (2010); Etaki et al. (2013). Alternatively, carbon nanotubes could be employed (acting as superconducting junctions in situ) to produce mechanical vibrations Schneider et al. (2012); Shevchuk et al. (2017). The NAMR (with mass and frequency ) vibrates along the -direction, and a time-dependent magnetic field is perpendicularly applied to the arm of the NAMR in the -direction Etaki et al. (2008); Poot et al. (2010).

Denoting the gauge-invariant phase difference by and the Josephson energy by for the th junction, we can write the total Josephson energy for the SQUID as ():

(1)

where is the total Josephson energy, is the junction asymmetry, and represents the overall phase of the SQUID (Schneider et al., 2012). Note that is bound by the fluxoid quantization relation, where is the flux through the SQUID ring and is the flux quantum. Here we have neglected the geometric and kinetic inductances of the loop Schwarz (2015). As shown in Fig. 1, there are two components for the flux  Xue et al. (2007a); Shevchuk et al. (2017): The static part , which is induced by a homogeneous magnetic field in the -direction, and a time-dependent part resulting from the NAMR vibrating around its equilibrium position and the -direction magnetic field . Therefore is expressed as

(2)

where is the average geometric constant Xue et al. (2007b), and is the displacement of the NAMR away from its equilibrium position at .

Note that and are induced by two magnetic fields with perpendicular directions. The first and second terms in Eq. (2) can, in principle, be changed independently. An imperfect perpendicular relation between these two magnetic fields might cause the net control flux, which should be minimized in experiments. To obtain the coupling relation between the vibration mode and the SQUID, we rewrite Eq. (1) as

(3)

where is the effective Josephson energy, and is the shifted phase. We assume that the junction asymmetry , and is far away from . As a result, is only a small constant phase factor, which has a small effect on the kinetic energy (Shevchuk et al., 2017). The vibrational motion of the NAMR induces a flux perturbation on . By expanding the displacement-dependent to first order in , we obtain

(4)

Considering that two junctions are symmetric with , we reduce Eq. (4) to a simpler form

(5)

from which we find that the magnetic field , together with the mechanical oscillations, induce a time-dependent modulation of the effective Josephson energy of the SQUID.

ii.2 Longitudinal interaction between superconducting qubits and NAMRs

Figure 2: (a) Schematic diagram for coupling a NAMR with a flux qubit. The small -junction for the flux qubit is replaced by the SQUID in Fig. 1. The NAMR (the red line) vibrates in the direction perpendicular to this plane. (b) Schematic diagram of a NAMR coupled to a transmon qubit. Similar to the flux qubit case (a), the single junction is replaced by the hybrid mechanical-SQUID system. The time-dependent flux changes the effective Josephson energy of the transmon qubit. In panels (a) and (b), the coherent driving fields are applied through the 1D transmission line (shown in green).

In typical circuit layouts of a superconducting qubit, we often replace a nonlinear Josephson junction with a SQUID for tuning the Josephson energy, which enables more flexibility and tunability for controlling qubits. For example, as shown in Fig. 2(a), we consider a gradiometric gap-tunable flux qubit, where the SQUID and two identical junctions with their Josephson energy , form two symmetric gradiometric loops Paauw et al. (2009); Paauw (2009); Fedorov et al. (2010); Stern et al. (2014). Compared with a three-junction flux qubit Mooij et al. (1999); Orlando et al. (1999), the small -junction is replaced by a mechanical-SQUID system, where is operated in the regime  Paauw (2009); Schwarz (2015); Orlando et al. (1999). The NAMR is shown by the red line. As demonstrated in Refs. Paauw et al. (2009); Fedorov et al. (2010), around the optimal point, the Hamiltonian for the flux qubit can be approximately expressed as

(6)

where and are Pauli operators, given in terms of the ground and excited states of the qubit. Moreover, the energy bias is controlled via the imbalance flux through the two gradiometric loops, and is the persistent-current amplitude. In experiments, can be induced via a propagating microwave field in a 1D transmission line [green line in Fig. 2(a)].

The energy gap depends on the parameter . Specifically, the following approximate analytical expression can be obtained by tight-binding calculations of the eigenstates Schwarz (2015); Orlando et al. (1999):

(7)

where . Note that linearly depends on the effective Josephson energy of the SQUID. Therefore, we can express the sensitivity of the gap to the flux control through the SQUID loop as follows Paauw (2009); Schwarz (2015)

(8)

where is obtained from Eq. (7), and is given in Eq. (4). Indeed, the flux sensitivity could be directly obtained in experiments by detecting a qubit spectrum via sweeping the flux control . As shown in Refs. Paauw (2009); Paauw et al. (2009), is about . The flux perturbation results from the vibrations of the NAMR and the time-dependent magnetic field  Wang et al. (2017b), and therefore, can be approximately rewritten as

(9)

Following Eqs. (6)-(9) and considering the NAMR free energy, the Hamiltonian of this hybrid system becomes

(10)

where () is the phonon annihilation (creation) operator of the NAMR. By expanding with its Fourier transform of frequency , the longitudinal-interaction strength can be written as

(11)

where is the zero-point fluctuation of the NAMR, and is the magnetic-field amplitude of the frequency component. However, the magnetic noise through the SQUID loop also causes decoherence of the qubit via the flux sensitivity in situ Paauw (2009); Schwarz (2015); Paauw et al. (2009), the relaxation times (both and ) decrease with increasing . To suppress these decoherence processes, we should control below a certain level.

Besides employing a flux qubit, it is possible to induce the time-dependent interaction between a transmon qubit Koch et al. (2007); You et al. (2007); Schreier et al. (2008); Mallet et al. (2009) and a NAMR. As depicted in Fig. 2(b), the single Josephson junction for the transmon qubit is replaced by a SQUID embedded in a NAMR. The charging energy is reduced by adding a large shunt capacitance  Koch et al. (2007); You et al. (2007); Schreier et al. (2008), and the transmon qubit is insensitive to charge noise under the condition . For simplicity, we assume that the driving field propagating along the 1D transmission line (plotted as green lines in Fig. 2) and the bias voltage are applied to the gate capacitance . Given that the transmon qubit, shown in Fig. 2(b), can be approximately viewed as a Duffing oscillator, the transition frequency between the two lowest eigenstates is Schneider et al. (2012); Koch et al. (2007):

(12)

For simplicity, we consider the symmetric junction at , then the flux sensitivity on of the transmon qubit is

(13)

By assuming that , , and , we obtain . Similar to the above discussions about our derivation of Eq. (12), we can also obtain the modulated coupling between the transmon qubit and the NAMR.

Here we discuss the coupling strength under current experimental conditions, and choose a carbon-nanotube resonator as the NAMR, e.g., with effective mass and fundamental frequency  Yao et al. (2000); Hüttel et al. (2009); Laird et al. (2011); Benyamini et al. (2014). The NAMR length can be of with geometric constant  Xue et al. (2007b); Etaki et al. (2008); Shevchuk et al. (2017). Employing the flux sensitivity and the magnetic field amplitude , the coupling strength for the frequency component is about .

There are many potential applications for such time-dependent longitudinally-coupled systems, such as generating macroscopic nonclassical states and performing quantum nondemolition measurements of the qubit states Didier et al. (2015); Royer et al. (2017).

In the following, we discuss the optical response of a microwave field applied to the qubit, and show how to observe various types of tunable EIT based on such a hybrid system.

Iii Effective Hamiltonian for tunable sideband transitions

To observe sideband transitions and EIT, we assume that a strong drive and a weak probe, with frequencies and , respectively, are applied to the superconducting qubit through the 1D transmission line Astafiev et al. (2010); Anisimov et al. (2011); Hoi et al. (2011, 2012). The Hamiltonian of the driven hybrid system can be written as

(14)

where and are the Rabi frequencies of the drive and probe fields, respectively, and () is the raising (lowering) operator for the qubit. Applying a frame rotating at frequency and adopting the rotating wave approximation, the Hamiltonian in Eq. (14) is transformed to

(15)

where is the qubit-drive detuning, is the probe-drive detuning, and . Assuming that , we can neglect the last term in Eq. (15). By applying the time-dependent polariton transformation to (see, e.g., Royer et al. (2017)), where we obtain the transformed Hamiltonian

(16)

where We can eliminate the longitudinal-coupling terms in Eq. (16) by setting . Assuming that , the following relations should be satisfied

(17)
(18)

To simplify our analysis, we assume that is sinusoidally modulated by a monochromatic drive at frequency , i.e.,

(19)

The general solution for the differential Eqs. (17) and (18) has the form

(20)

where is an arbitrary complex coefficient. For simplicity, by setting , is reduced to

(21)

Under the condition , Eq. (16) is simplified as

(22)

The coupling-modulation frequency and the coupling constant are assumed to be much smaller than the NAMR frequency, i.e., . Therefore, is always a small dimensionless parameter, as it holds

(23)

Expanding the last term of Eq. (22) to third order in , the Hamiltonian reads

(24)

We assume a red-sideband drive field with an amplitude, which is weak compared with the detuning frequency, i.e., . Similar to discussions in Ref. Wang et al. (2016), the third term in Eq. (24) leads to a dynamical Stark shift for the qubit. We can rewrite the Hamiltonian in Eq. (24) in the basis of its eigenstates:

(25a)
(25b)

with . Since , we neglect the rotating angle of the qubit basis. The dynamical Stark shift slightly shifts the qubit-transition frequency. Thus, we should replace the detuning in the qubit free-energy term in Eq. (24) with a modified qubit-drive detuning

(26)

We assume that the drive detuning , is of the same order as coupling-modulation frequency . Then by applying the unitary transformation to Eq. (24) and by neglecting rapidly-oscillating terms, we obtain the effective Hamiltonian

(27)

Note that has been decomposed into the frequency components and as follows:

(28)

with the first- and third-order sideband transition rates given respectively by:

(29)
(30)

where . The NAMR is assumed to be in the quantum regime with several phonons, and is not large. When deriving Eq. (27), the rapidly-oscillating terms were neglected and only near-resonant ones are kept. Equation (24) was expanded to third order in only, as observable effects of higher-order terms can be ignored. If sweeping the sideband drive frequency around the regime , four apparent resonant positions can be observed at . It is easy to find first-order transition rates

(31)

by assuming that . In some of the following discussions, we neglect the third-order transition rates . To observe the optical response of the probe field, the probe term should be added (in the rotating frame), then the total Hamiltonian becomes

(32)
Figure 3: Lowest-energy levels of the qubit-NAMR system and possible state transitions induced by a sideband drive (of frequency , which is not shown here) and a resonant probe (of frequency , represented by the red arrow). The dynamical Stark shift results from the detuning of the drive field. Due to a sinusoidal modulation of the longitudinal coupling at frequency , the first-order sideband transition, induced by the drive field (from to ), is split into two transitions with rates (blue arrows) and (green arrows). Note that () is the qubit (NAMR) frequency, and is the coupling-modulation frequency.

The energy-level diagram is depicted in Fig. 3. Due to the sinusoidal modulation of the longitudinal coupling, the monochromatic drive field with strength in the original Hamiltonian, given by Eq. (14), induces two coherent transition processes between the states . The corresponding transition rates are (as shown with blue and green arrows). After compensating the dynamical Stark shift , the frequency separation between these two coherent transitions is equal to the doubled coupling-modulation frequency, i.e., . In the following discussions, we will show that, assuming that the drive field is tuned properly, both conventional and bichromatic EIT (or two-color EIT) can be observed in such a hybrid system.

Iv Electromagnetically induced transparency

In this section, we discuss the optical response to the probe field via the standard master equation approach in our proposal. The whole system is assumed to be coupled to a zero-temperature environment, and the Born-Markov approximation is valid here. Therefore, the system evolution is approximately described by the Lindblad-type master equation

(33)

where are the decoherence terms of the Lindblad superoperator form, () is the decay (pure dephasing) rate of the qubit, and is the relaxation rate of the mechanical mode due to its coupling environment with a mean thermal phonon number . Employing the methods of Refs. Astafiev et al. (2010); Abdumalikov et al. (2010); Hoi et al. (2011, 2012), one can approximately assume that the decay of the qubit is caused only by the quantum noise in a 1D open line. The coherent drive and probe fields for the qubit are also applied through the 1D transmission line. Since the size of the qubit loop () is much smaller than the wavelength of the microwave drive (), we assume that the drive is place-independent Astafiev et al. (2010).

The time-dependent atomic dipole moments can be expanded in terms of the frequency Fourier components as Yan et al. (2013)

(34)

where can be found by numerically solving the master equation (33). Different from employing a susceptibility to describe the optical response of an atomic ensemble Scully and Zubairy (1997); Fleischhauer et al. (2005), here we should use the reflection coefficient to characterize the electromagnetic response for the probe field of a single atom Wang et al. (2017b), which can be obtained via the following relation Astafiev et al. (2010); Abdumalikov et al. (2010)

(35)

The real and imaginary parts of the reflection coefficient are related to the reflection and dispersion of a single atom, respectively. In the following, we discuss how behaves under different drive and probe conditions.

iv.1 Single-color EIT

Figure 4: Single-color EIT with . (a) The real and imaginary parts of the reflection coefficient , (red solid curve) and (blue solid curve), as functions of the drive-probe detuning based on our numerical simulation. The curves shown with stars and circles are plotted according to analytical Eq. (40). At , displays a dip with . (b) Time evolutions of the dark-state fidelity , given in Eq. (33), and of the qubit for the dip position in (a). Note that oscillates around zero. (c) Employing the fast Fourier transform, in (b) is decomposed in the frequency domain. Since our numerical calculations are performed in the rotating frame of the probe field frequency, the dc (zero frequency) component of corresponds to the optical response of the probe field, which has a low amplitude with , when EIT occurs. The parameters are: , , , , , , , , and .

The two split-sideband transitions are well-separated under the condition . Assuming that (), only the sideband transition () is on resonance, and we reduce the Hamiltonian, in Eq. (32), by adopting the rotating wave approximation, as follows

(36)

Under these conditions, the dynamical Stark shifts of the qubit for the Hamiltonians are expressed as

(37)

To observe single-color EIT, we assume that the sideband transition is resonantly selected. When the probe-drive detuning satisfies the condition , as shown in Fig. 3, the transitions are represented by the red and blue arrows in Fig. 3. As a result, the evolution of the system is approximately described by the time-independent Hamiltonian

(38)

Assuming that the NAMR has a high-quality factor and the condition is satisfied, the decoherence process of the NAMR can be neglected. The effective Hamiltonian , together with the rapid decay of the qubit, drives the system into the following dark state Sun et al. (2006); Wang et al. (2016):

(39)

where and is a coherent state.

In a typical EIT system, the probe field is weak compared with the control field, i.e., . Therefore , and we can use the states , , and to describe the transitions governed by Eq. (36). The relation between these three states is similar to a -type EIT system. The reflection coefficient of the probe field for in Eq (36) is expressed as Scully and Zubairy (1997); Fleischhauer et al. (2005):

(40)

where is the total dephasing rate. Equation (40) indicates that the mechanical decay rate determines the width of the EIT window. Thus, very narrow EIT windows can be observed in our proposal by adopting a high quality-factor NAMR with . We assume that all the parameters are set to be dimensionless: the mechanical frequency is , and the Rabi frequencies of the two coherent drives are and . For the modulated coupling , we set and . According to Eq. (29), the effective sideband transition rate is . Employing these parameters, in Fig. 4(a) we plot (red solid curve) and (blue solid curve) changing with the detuning by numerically solving the master equation with the original Hamiltonian in Eq. (15) (rotating at the probe frequency). Moreover, the analytical form for , given in Eq. (40), is also plotted with the curves shown with symbols.

In Fig. 4(a), one can see a single EIT dip with around . Different from conventional atomic EIT, the control field here is not a semiclassical coherent drive, but a parametrically modulated coupling inducing a sideband transition . Moreover, we find that our analytical and numerically results of the optical response match well with each other, indicating that the Hamiltonian Eq. (38) can effectively describe the transition relation of the single-window EIT in Fig. 4(a).

Defining the fidelity

(41)

for the dark state in Eq. (39), Fig. 4(b) depicts the time evolution of this fidelity and with [the dip position in Fig. 4(a)]. The numerical results clearly show that the system is rapidly steered into its dark state with the steady fidelity . Note that oscillates in time, and it contains many frequency components. The fast Fourier transform of is shown in Fig. 4(c), which exhibits three main peaks in the low-frequency regime. The first peak at (i.e., the dc component) corresponds to the optical response of the probe field. This peak has very low amplitude due to EIT.

Figure 5: (a) The reflection rate of the resonant probe field versus the drive detuning and the coupling-modulation frequency . At , there is only one sideband drive frequency for the single-color EIT window. With increasing , the single dip for the EIT drive field is split into two apparent dips at , and two less apparent dips at as additionally indicated by the dotted red lines. (b) The cross-section of figure (a) at . Two shallow dips induced by the third-order terms can be observed at . Other parameters employed here are the same as those in Fig. 4.

The single-color EIT demonstrated here is quite different from the case when the qubit-NAMR longitudinal coupling is constant Li and Zhu (2012); Wang et al. (2016), where the drive-probe detuning should be fixed and exactly equal to the eigenfrequency of the NAMR. In the case studied here with a modulation interaction, the drive-probe detuning is continuously changed by varying the coupling-modulation frequency , which is analogous to changing the frequency difference between two metastable states in the EIT system. This can be clearly seen in Fig. 5(a), where we fix the probe field to be resonantly applied to the qubit, and plot the optical response by changing the drive detuning and the coupling-modulation frequency . At , there is only one drive-probe detuning position for the single EIT window (). Intriguingly, when we start to increase the frequency from zero, the single dip of the reflection rate as a function of the drive detuning splits into two apparent dips separated by , which corresponds to and , respectively. Two possible drive frequencies can induce single-color EIT. Thus, modulating the qubit-NAMR coupling at frequency is equivalent to replacing the single NAMR with two frequency tunable NAMRs with frequencies .

Moreover, we find two shallow dips along the dashed lines in Fig. 5(a), which can be seen clearly in the cross-section plot Fig. 5(b) by fixing the coupling-modulation frequency at . The relation between drive detuning and the modulation frequency is approximately given by . These two transparent dips result from the third-order resonant couplings in Eq. (30). However, due to extremely low rates, these two dips are much shallower than those induced by .

iv.2 Two-color EIT

In previous discussions, we found that only one sideband transition was dominant by setting the coupling-modulation frequency . However, by assuming that , the two sideband transitions (with strengths ) correspond to the same detuning, so both of these should be considered equally. The Hamiltonian in Eq. (32) is now reduced to

(42)

As shown in Fig. 3, both sideband transitions and are detuned by . The energy-level transition relation is similar to two-color EIT with two control fields studied in Refs. Wang et al. (2006); Moiseev and Ham (2006); Yan et al. (2013). Two transparent windows for the probe field were observed. In contrast to two-color EIT studies in atomic systems Wang et al. (2003); Moiseev and Ham (2006); Liu et al. (2012); Yan et al. (2013); Liu et al. (2012), only one (not two) coherent drive is employed here. The two split transparent windows result from a monochromatic modulation of the longitudinal coupling. Indeed, if contains well-separated frequency components, windows of EIT can be observed.

Figure 6: Two-color EIT under the condition . The real and imaginary parts of the reflection coefficient , (red solid curve) and (blue solid curve), versus the drive-probe detuning based on our numerical simulations. The analytical average optical response , as given in Eq. (44), is plotted with symbols. Two transparent dips can be found at . Other parameters adopted here are the same as those in Fig. 4.

Analogous to a conventional -type EIT system, if we consider only one detuning sideband transition (either or ) of detuning in Eq. (42), the reflection coefficient becomes

(43)

with a real part, which has dip positions at . If both sideband transitions and occur with a symmetric detuning, the optical response for the probe field should combine these two EIT effects.

In Fig. 6, the imaginary and real parts of the reflection coefficient are plotted based on our numerical simulations (the blue and red solid curves). Around , the two EIT windows emerge with typical anomalous dispersion curves of negative slope. Therefore, by applying a single-drive field , we can simultaneously control the transparency for two microwave fields when their frequency separation equals . Moreover, following Eq. (43), the analytical mean optical response can be approximately expressed as

(44)

which is plotted by the curves with symbols (either stars or circles) in Fig. 6. Interestingly, we find that the analytical results in Eq. (44) can approximately describe the joint two-color EIT. The optical response can be viewed as a combined effect of two isolated EIT effects with the same drive detuning, and their interference is negligible, given that are much weaker compared with . As shown in Eq. (31), the sideband transition is greater than , and the rate difference becomes more apparent when increasing the modulation frequency . When is large enough, the two dips in Fig. 6 are not symmetric anymore.

Figure 7: The real part of the reflection coefficient, , versus the drive-probe detuning and drive strength for the two-color EIT. When increasing , the two EIT dips located at , become wider and closer to each other, indicating that the two-color EIT is gradually destroyed by the interference of the two sideband transitions. Other parameters adopted here are the same as those in Fig. 4.

According to Eq. (29), both and linearly increase with increasing . In Fig. 7, we plot the reflection rate as a function of the drive strength and probe detuning . We find that, with increasing , the two EIT dips become wider and closer due to strong sideband transition rates . The two isolated transparent windows affect each other, and they tend to merge. When is comparable with the frequency separation , the relation is not valid any more, and this two-color EIT almost disappears. If the two sideband transitions are not well-separated by frequency detuning, Eq. (44) cannot effectively describe the optical response, and the two EIT windows are destroyed.

V Discussion and conclusions

In this work, we considered a hybrid system consisting of a SQUID embedded with a NAMR. We first showed an example of how to achieve an unconventional parametrically-modulated longitudinal interaction between a flux (transmon) qubit and the NAMR. Then, we derived an effective Hamiltonian, which leads to a first-order sideband transition, and found that the coupling modulation significantly changes the dynamics of the hybrid system. A single sideband drive is split under a sinusoidal modulation of the coupling terms. Indeed, the frequency components of the modulation directly determine this splitting. If the modulation is more complex, then more interesting phenomena can be observed.

By applying a resonant probe field, we found that both single- and two-color EIT can be observed. The modulation of the interaction provides another control method for these EIT effects. For the single-color EIT, the drive-probe detuning is not necessarily equal to the NAMR frequency, but can be conveniently tuned by changing the modulation frequency. For the two-color EIT, the double transparent windows occur due to the splitting of sideband transitions, and their distance is determined by the modulation frequency. Compared with the usual predictions of two-color EIT in atomic systems, here there is only one drive (control) field. Moreover, it is possible to modify and extend our results to study EIT and Autler-Townes splitting Anisimov et al. (2011); Peng et al. (2014); Sun et al. (2014).

As discussed in Ref. Wang et al. (2017b), for systems with longitudinal interaction, an EIT induced by second-order sideband transitions can also be observed. By considering the modulation of such longitudinal interaction, one might observe multi-color EIT induced by higher-order sideband transitions.

We hope that our results could not only be helpful for studying the dynamics for a system with time-dependent longitudinal coupling, but also can find applications in microwave photonics Gu et al. (2017) (including vacuum-induced nonlinear optics Kockum et al. (2017); Stassi et al. (2017)) and quantum information processing Wendin (2017) with SQCs.

Acknowledgements

X.W. and H.R.L. were supported by the Natural Science Foundation of China under Grant No. 11774284. A.M. and F.N. acknowledge the support of a grant from the John Templeton Foundation. F.N. is partially supported by the MURI Center for Dynamic Magneto-Optics via the AFOSR Award No. FA9550-14-1-0040, the Japan Society for the Promotion of Science (KAKENHI), the IMPACT program of JST, CREST Grant No. JPMJCR1676, RIKEN-AIST Challenge Research Fund, JSPS-RFBR Grant No. 17-52-50023.

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