Phase control of electromagnetically induced acoustic wave transparency in a diamond nanomechanical resonator

Physics Letters A 381 (2017) 1624–1628

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Physics Letters A www.elsevier.com/locate/pla

Phase control of electromagnetically induced acoustic wave transparency in a diamond nanomechanical resonator Sofia Evangelou Materials Science Department, School of Natural Sciences, University of Patras, Patras 265 04, Greece

a r t i c l e

i n f o

Article history: Received 27 January 2017 Received in revised form 5 March 2017 Accepted 7 March 2017 Communicated by V.A. Markel Keywords: Nitrogen-vacancy center in diamond Diamond nanomechanical resonator Acoustic field Microwave field Phase control Transparency

a b s t r a c t We consider a high-Q single-crystal diamond nanomechanical resonator embedded with nitrogenvacancy (NV) centers. We study the interaction of the transitions of the spin states of the ground state triplet of the NV centers with a strain field and two microwave fields in a -type coupling configuration. We use the relative phase of the fields for the control of the absorption and dispersion properties of the acoustic wave field. Specifically, we show that by changing the relative phase of the fields, the acoustic field may exhibit absorption, transparency, gain and very interesting dispersive properties. © 2017 Elsevier B.V. All rights reserved.

1. Introduction When a quantum system interacts with electromagnetic fields in a closed loop coupling configuration, then the optical properties of the system depends strongly on the phases of the fields [1–13]. This leads, for example, to the change of the behavior of a probe field from absorption to optical transparency and to even gain simply by altering the phases of the applied fields. Similar phenomena are also obtained when the quantum system exhibits vacuum induced coherence and interacts with external fields with varying phases [14–18]. The prototype quantum system for phasedependent optical effects is the three-level -type system, where each transition of the system interacts with a different field. Recently, an interesting variation of electromagnetically induced transparency [19] has been proposed in a high- Q single-crystal diamond nanomechanical resonator embedded with nitrogenvacancy (NV) centers [20]. By using the transitions of the spin states of the ground state triplet of the NV centers under the interaction with strain and microwave fields, in either  or -type coupling configurations, Hou et al. [20] showed that the system may exhibit transparency for an acoustic wave field; a phenomenon that was termed electromagnetically induced acoustic wave transparency. Here, we propose an extension of this idea by controlling the absorption and dispersion properties of the acous-

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.physleta.2017.03.008 0375-9601/© 2017 Elsevier B.V. All rights reserved.

tic wave field by the use of the relative phases of the fields for the  coupling configuration. We concentrate to the case of one weak microwave field coupling the lower state to the middle state and one weak to strong microwave field coupling the two upper states, and show that by changing the relative phase of the fields, the acoustic field may exhibit absorption, transparency, gain and even “slow sound”, as well as, negative group velocities. We also present specific conditions for complete transparency, gain and zero absorption with non-zero dispersion. We note that phase-dependent optical absorption-gain and optical switching of NV centers has been studied by Li et al. [12] using two optical fields and a microwave field and coupling the ground state with excited states of the NV center in a -type coupling configuration. Electromagnetically induced transparency under -type configurations using optical and microwave fields to couple the ground and excited states of the NV has been experimentally observed and theoretically analyzed by Manson and coworkers [21]. Also, recently Hou et al. [22] proposed the generation of macroscopic Schrödinger cat states in a high- Q single-crystal diamond nanomechanical resonator embedded with NV centers and using a -type coupling configuration of the NV centers ground triplet by combining a quantized mechanical strain field with two microwave fields. Moreover, significant experimental progress has been made in the coherent coupling of the ground state triplet, as well as excited states, of NV centers, including diamond mechanical resonators, with strain fields and electromagnetic fields [23–32]. In these works a -type coupling configuration of the

S. Evangelou / Physics Letters A 381 (2017) 1624–1628

ground state triplet by two magnetic fields and an acoustic field has been already considered by Fuchs and co-workers [26], who also discussed the potential for phase control of the properties of the system. Also, Wang and co-workers [31] used the phase difference between an acoustic field and two optical fields, coupling the ground and excited states of a NV center, for observing coherent control of its fluorescence. This paper is organized as follows. In the next section we present the model, the theory and the main results. Our findings are summarized in section 3. 2. Theoretical model and results The system that we consider is based on a high- Q singlecrystal diamond nanomechanical resonator embedded with many NV centers. The quantum system is based on the ground electron spin states of the NV center. The electronic structure of the NV center ground state and the interaction with electric, magnetic and strain fields has been discussed in detail in several papers, see for example, refs. [33,34]. We take each NV center to be negatively charged with two unpaired electrons located at the vacancy. Then, the ground state has a spin-triplet form. The corresponding levels for the states of the ground state |ms  (with ms = 0, ±1) of the NV center have a zero-field splitting of about 2.87 GHz between |0 and |±1 [34]. The application of an external moderate magnetic field lifts the degeneracy and splits the states |±1. It also shifts the |−1 state to be below state |0 [35,36]. The corresponding three-level system is shown in Fig. 1. The transition between states |−1 and |+1 is electric dipole forbidden. However, the two states can be coupled coherently with a strain field [20,22–32,37]. The transitions from |−1 to |0 and from |0 to |+1 are electric dipole allowed. These two transitions are driven by two microwave fields. The Hamiltonian which describes the coupling of the above system with a strain field p, which is described by a local electric field with frequency ω p , amplitude E p and phase ϕ p , as well as with two microwave fields denoted a and b, in the interaction picture and the rotating wave approximation, is given by:

H=





¯ k |kk| − h¯  p e −i ω p t −i ϕ p |+1−1| h¯ ω

k=0,±1

 + a e −i ωa t −i ϕa |0−1| + b e −i ωb t −i ϕb |+10| + H.c. . (1)

¯ k is the energy of each state, ωn and ϕn , with n = a, b, Here, h¯ ω are the frequency and the phase of the microwave fields a and b, respectively. Also, m , with m = p , a, b, is the Rabi frequency for each one of the three transitions (taken real for simplicity). We study the absorption/dispersion properties of a weak strain field using a probability amplitude approach. This approach gives correct results for weak fields p and a. In this respect, the

Fig. 1. The level scheme for the NV center ground state sublevels and the relevant couplings. States |−1 and |+1 are coupled by a strain field. The transitions from |−1 to |0 and |0 to |+1 are driven by two microwave fields a and b, respectively.

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wavevector of the system, at a specific time t, can be expanded in terms of the ‘bare’ eigenvectors such that

|ψ(t ) = c −1 (t )e −i ω¯ −1 t | − 1 + c 0 (t )e −i ωa t −i ω¯ −1 t −i ϕa |0 + c +1 (t )e −i ω p t −i ω¯ −1 t −i ϕ p | + 1 .

(2)

Substituting Eqs. (1) and (2) into the time-dependent Schrödinger equation we obtain the time evolution of the probability amplitudes as

i c˙ −1 (t ) = − p c +1 (t ) − a c 0 (t ) ,

(3) iϕ

i c˙ 0 (t ) = − (δ − b + i γ0 ) c 0 (t ) − a c −1 (t ) − b e c +1 (t ) , i c˙ +1 (t ) = − (δ + i γ1 ) c +1 (t ) −  p c −1 (t ) − b e

−i ϕ

c 0 (t ) ,

(4) (5)

¯ +1 + ω¯ −1 and b = ωb − with the detunings defined as δ = ω p − ω ω¯ +1 + ω¯ 0 . Above, we have assumed that ω p = ωa + ωb . Also, ϕ = ϕa + ϕb − ϕ p is the phase difference between the microwave fields and the strain field. Finally, γ0 and γ1 denote the decay rates of states |0 and |+1, respectively. The steady state linear susceptibility for the acoustic wave field is given by [20,38,39]

χ (ω p ) ∝ N

∗ (t → ∞)c (t → ∞) c− +1 1

p

(6)

,

where N is the density of the NV centers. We take that the NV centers are initially in state |−1. Then, the initial conditions of the probability amplitudes are c −1 (t = 0) = 1, c 0 (t = 0) = 0, and c +1 (t = 0) = 0. We assume that the interaction of the NV centers with the strain field and the microwave field a is very weak so that c −1 (t ) ≈ 1 for all times. Then, by using perturbation theory, and under a steady state, we obtain c +1 (t → ∞) from Eqs. (3)–(5), so the steady state susceptibility for the acoustic wave field takes the form

χ (ω p ) ∝ N

−(δ − b + i γ0 ) +

a −i ϕ  p b e

(δ − b + i γ0 )(δ + i γ1 ) − b2

.

(7)

We note that Eq. (7) can also be obtained by a density matrix approach and using a first order expansion in terms of fields p and a. Eq. (7) shows that the susceptibility depends on several parameters namely a / p , ϕ , b , b . Here, we will use the phase difference ϕ for obtaining phase control of the absorption-gain and dispersion properties of the acoustic field. In Fig. 2 we present the real and imaginary part of the acoustic field susceptibility as a function of the detuning δ for different values of the phase difference ϕ . In Fig. 2(a), which is for ϕ = 0, we find that the absorption spectrum becomes zero at two values of δ and between these values Im(χ ) < 0 showing gain. Also, at one of the two values of the detunings, the larger positive value, Re(χ ) = 0 as well, so at this value of ω p there is complete transparency. For ϕ = π /4 and ϕ = π /2, shown in Figs. 2(b) and 2(c), only absorption of the acoustic field is obtained. The absorption spectrum for ϕ = π /4 is asymmetric, with a strong peak at negative detunings and a very weak peak at positive detunings, while the absorption spectrum for ϕ = π /2 is symmetric with two peaks at symmetric values of the detuning and a weak dip at the exact resonance. For ϕ = π , shown in Fig. 2(d), the spectrum is essentially mirror image of the one for ϕ = 0 so one obtains complete transparency, this time for negative values of the detuning, and gain between the zero values of Im(χ ). For ϕ = 5π /4 and ϕ = 3π /2, shown in Figs. 2(e) and 2(f), the absorption spectrum shows two zeroes at specific values of the detuning and gain between these values. The zero values of Im(χ ) are close to the highest and lowest values of Re(χ ), which leads to enhancement of the

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S. Evangelou / Physics Letters A 381 (2017) 1624–1628

Fig. 2. Im(χ ) (solid curve) and Re(χ ) (dashed curve), in arbitrary units, as a function of δ for parameters b = γ1 , ϕ = π /4, (c) ϕ = π /2, (d) ϕ = π , (e) ϕ = 5π /4, and (f) ϕ = 3π /2.

γ0 = 0.01γ1 , b = 0, a / p = 1 and (a) ϕ = 0, (b)

index of refraction without absorption [40,41]. Also, for ϕ = 5π /4 the absorption spectrum is asymmetric and for ϕ = 3π /2 the absorption spectrum is symmetric. We will now analyze the case that all fields are at exact reso¯ +1 − nance δ = b = 0. For the acoustic field this means ω p = ω ω¯ −1 = ω+1,−1 . In this case the susceptibility becomes:

χ (ω p = ω+1,−1 ) ∝ N

 a −  cos ϕ + i γ0 + b p

a  p b sin ϕ

γ0 γ1 + b2

 . (8)

We first present the conditions for exact transparency at ω p = ω+1,−1 (Im[χ (ω+1,−1 )] = 0 and Re[χ (ω+1,−1 )] = 0) which are

a γ0 = p b

,

π ϕ = (2m + 1) , m = 1, 3, 5, .... 2

(9)

Then, we present the conditions corresponding to zero absorption (Im[χ (ω+1,−1 )] = 0) but non-zero dispersion (Re[χ (ω+1,−1 )] = 0), which leads to enhancement of the index of refraction without absorption [40,41]. These conditions are

a γ0 sin ϕ = − p b

,

π

ϕ = (2m + 1) , m = 1, 3, 5, .... 2

(10)

Then,

χ (ω p = ω+1,−1 ) ∝ − N

a  p b cos ϕ

γ0 γ1 + b2

.

(11)

Fig. 3. Im(χ ) (solid curve) and Re(χ ) (dashed curve), in arbitrary units, at ω p = ω+1,−1 as a function of ϕ for parameters b = 0.01γ1 , γ0 = 0.01γ1 , b = 0, and a / p = 1.

For γ0 b Eq. (10) can be approximately fulfilled for ϕ ≈ mπ , with m = 0, 1, 2, ..., which also gives the minimum and maximum values of Re(χ ). Finally, we present the conditions for gain at ω p = ω+1,−1 (Im[χ (ω+1,−1 )] < 0) which reads

a γ0 sin ϕ < − , p b

(12)

which could be fulfilled either for specific regions of ϕ or for specific regions of a / p . For example, for γ0 b Eq. (12) can be approximately fulfilled for sin ϕ < 0, so it is fulfilled for ϕ ∈ [(2m + 1)π , 2(m + 1)π ], with m = 0, 1, 2, .... In Fig. 3 we present the real and imaginary part of the acoustic field susceptibility at ω p = ω+1,−1 as a function of the phase

S. Evangelou / Physics Letters A 381 (2017) 1624–1628

Fig. 4. Im(χ ) (solid curve) and Re(χ ) (dashed curve), in arbitrary units, at ω p = ω+1,−1 as a function of ϕ for parameters γ0 = 0.01γ1 , b = 0, a / p = 1, and (a) b = 0.05γ1 , (b) b = γ1 .

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Fig. 5. (a) Im(χ ) and (b) ∂ Re(χ )/∂ ω p , in arbitrary units, as a function of δ for parameters γ0 = 0.01γ1 , b = 0, a / p = 1, b = 0.05γ1 , and ϕ = 0 (solid curve), ϕ = π (dotted curve), ϕ = 5π /4 (dashed curve) and ϕ = 3π /2 (dot-dashed curve).

difference ϕ . As expected the behavior of both Im(χ ) and Re(χ ) γ0 a is periodic with ϕ , and as the relation  = is fulfilled for the p

b

chosen parameters, we obtain that both Re(χ ) and Im(χ ) becomes zero at ϕ = (2m + 1) π2 , with m = 1, 3, 5, .... Then, in Fig. 4 we present the real and imaginary part of the acoustic field susceptibility at ω p = ω+1,−1 as a function of the phase difference ϕ for different values of b that do not fulγ0 a =  . The behavior is again periodic and the zeroes of fill  p

b

Im(χ ) are obtained near ϕ = mπ , with m = 0, 1, 2, ..., while the minima and maxima of Re(χ ) is also obtained at ϕ = mπ , with m = 0, 1, 2, .... Therefore, enhancement of the index of refraction without (or with very little) absorption can be obtained for these values of ϕ . We also note that for smaller values of b the values of both Re(χ ) and Im(χ ) change over larger region of values. Also, in both Figs. 2 and 3 gain is obtained in the predicted regions. Interesting results may be obtained for Re(χ ) as well. The group velocity of the acoustic wave field is given by [20]

vg =

v 1 + (1/2)Re(χ ) + (ω p /2)(∂ Re(χ )/∂ ω p )

,

(13)

with v being the sound propagating speed in pure diamond. Here, the derivative of the real part of the susceptibility is evaluated at the carrier frequency. The ∂ Re(χ )/∂ ω p and thus the group velocity can be strongly changed by the phase ϕ . This is shown in Fig. 5. We note that ∂ Re(χ )/∂ ω p takes large values, both positive and negative, near resonance. The large positive value of ∂ Re(χ )/∂ ω p leads to “slow sound”, which is the acoustic analogue of slow light [42,43]. Also, the large negative values of ∂ Re(χ )/∂ ω p leads to negative values of group velocity of the acoustic field, which gives an acoustic analogue of superluminal light [43]. The large values of ∂ Re(χ )/∂ ω p is also accompanied with low values of Im(χ ). Therefore, they could be experimentally observable. We note that there are experimental observations of both “slow sound” [44,45] and negative group velocities for the acoustic field [46,47] in different structures. The strong phase dependence of ∂ Re(χ )/∂ ω p and the strong modulation of its value is also shown in Fig. 6, where we present ∂ Re(χ )/∂ ω p at ω p = ω+1,−1 as a function of the phase difference ϕ .

Fig. 6. ∂ Re(χ )/∂ ω p , in arbitrary units, at ω p = ω+1,−1 as a function of ϕ for parameters γ0 = 0.01γ1 , b = 0, a / p = 1, and (a) b = 0.01γ1 (dashed curve) and b = 0.05γ1 (solid curve).

3. Summary We consider a high- Q single-crystal diamond nanomechanical resonator embedded with nitrogen-vacancy (NV) centers. We use the transitions of the spin states of the ground state triplet of the NV centers under the interaction with a strain field and two microwave fields in a -type coupling configuration, similar to the work of Hou et al. [20]. The different issue of this proposal is the control of the absorption and dispersion properties of the acoustic wave field by the use of the relative phase of the fields. We concentrate to the case of one weak microwave field and another weak to strong microwave field and show that by changing the relative phase of the fields, the acoustic field may exhibit absorption, transparency, gain and even “slow sound”, as well as negative group velocities. We also present specific conditions for complete transparency, gain and zero absorption with non-zero dispersion. References [1] D.V. Kosachiov, B.G. Matisov, Y.V. Rozhdestvensky, J. Phys. B 25 (1992) 2473. [2] E.A. Korsunsky, N. Leinfellner, A. Huss, S. Baluschev, L. Windholz, Phys. Rev. A 59 (1999) 2302. [3] F. Ghafoor, S.Y. Zhu, M.S. Zubairy, Phys. Rev. A 62 (2000) 13811. [4] D. Bortman-Arbiv, A.D. Wilson-Gordon, H. Friedmann, Phys. Rev. A 63 (2001) 043818. [5] M. Sahrai, H. Tajalli, K.T. Kapale, M.S. Zubairy, Phys. Rev. A 70 (2004) 023813. [6] P. Král, I. Thanopulos, M. Shapiro, D. Cohen, Phys. Rev. Lett. 90 (2003) 033001. [7] J.F. Dynes, E. Paspalakis, Phys. Rev. B 73 (2006) 233305. [8] P. Král, I. Thanopulos, M. Shapiro, Rev. Mod. Phys. 79 (2007) 53.

1628

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

[24] [25] [26] [27] [28] [29]

S. Evangelou / Physics Letters A 381 (2017) 1624–1628

H. Sun, Y.-P. Niu, S. Jin, S.-Q. Gong, J. Phys. B 40 (2007) 3037. J. Joo, J. Bourassa, A. Blais, B.C. Sanders, Phys. Rev. Lett. 105 (2010) 073601. W.Z. Jia, L.F. Wei, Phys. Rev. A 82 (2010) 013808. J.H. Li, R. Yu, X. Yang, Appl. Phys. B 111 (2013) 65. G.-L. Cheng, Y.-P. Wang, W.-X. Zhong, A.-X. Chen, Ann. Phys. 353 (2015) 64. E. Paspalakis, P.L. Knight, Phys. Rev. Lett. 81 (1998) 293. S.-Q. Gong, E. Paspalakis, P.L. Knight, J. Mod. Opt. 45 (1998) 2433. M. Macovei, J. Evers, C.H. Keitel, Phys. Rev. Lett. 91 (2003) 233601. X. Fan, N. Cui, S. Tian, H. Ma, S.-Q. Gong, Z-Z. Xu, J. Mod. Opt. 52 (2005) 2759. E. Paspalakis, S. Evangelou, V. Yannopapas, A.F. Terzis, Phys. Rev. A 88 (2013) 053832. M. Fleischhauer, A. Imamoglu, J.P. Marangos, Rev. Mod. Phys. 77 (2005) 633. Q. Hou, W. Yang, C. Chen, Z. Yin, J. Opt. Soc. Am. B 33 (2016) 2242. E.A. Wilson, N.B. Manson, C. Wei, L. Yang, Phys. Rev. A 72 (2005) 063813. Q. Hou, W. Yang, C. Chen, Z. Yin, Sci. Rep. 6 (2016) 37542. For a recent review of coherent coupling of strain fields with NV centers see D. Lee, K.W. Lee, J.V. Cady, P. Ovartchaiyapong, A.C.B. Jayich, J. Opt. 19 (2017) 033001. P. Ovartchaiyapong, K.W. Lee, B.A. Myers, A.C.B. Jayich, Nat. Commun. 5 (2014) 4429. J. Teissier, A. Barfuss, P. Appel, E. Neu, P. Maletinsky, Phys. Rev. Lett. 113 (2014) 020503. E.R. MacQuarrie, T.A. Gosavi, A.M. Moehle, N.R. Jungwirth, S.A. Bhave, G.D. Fuchs, Optica 2 (2015) 233. E.R. MacQuarrie, T.A. Gosavi, S.A. Bhave, G.D. Fuchs, Phys. Rev. B 92 (2015) 224419. A. Barfuss, J. Teissier, E. Neu, A. Nunnenkamp, P. Maletinsky, Nat. Phys. 11 (2015) 820. S. Meesala, Y.-I. Sohn, H.A. Atikian, S. Kim, M.J. Burek, J.T. Choy, M. Loncar, Rev. Phys. Appl. 5 (2016) 034010.

[30] K.W. Lee, D. Lee, P. Ovartchaiyapong, J. Minguzzi, J.R. Maze, A.C.B. Jayich, Rev. Phys. Appl. 6 (2016) 034005. [31] D.A. Golter, T. Oo, M. Amezcua, K.A. Stewart, H. Wang, Phys. Rev. Lett. 116 (2016) 143602. [32] D.A. Golter, T. Oo, M. Amezcua, I. Lekavicius, K.A. Stewart, H. Wang, Phys. Rev. X 6 (2016) 041060. [33] J.R. Maze, A. Gali, E. Togan, Y. Chu, A. Trifonov, E. Kaxiras, M.D. Lukin, New J. Phys. 13 (2011) 025025. [34] M.W. Doherty, F. Dolde, H. Fedder, F. Jelezko, J. Wrachtrup, N.B. Manson, L.C.L. Hollenberg, Phys. Rev. B 85 (2012) 205203. [35] L. Jin, M. Pfender, N. Aslam, P. Neumann, S. Yang, J. Wrachtrup, R.-B. Liu, Nat. Commun. 6 (2015) 8251. [36] S. Putz, A. Angerer, D.O. Krimer, R. Glattauer, W.J. Munro, S. Rotter, J. Schmiedmayer, J. Majer, Nat. Photonics 11 (2017) 36. [37] S.D. Bennett, N.Y. Yao, J. Otterbach, P. Zoller, P. Rabl, M.D. Lukin, Phys. Rev. Lett. 110 (2013) 156402. [38] A.V. Shvetsov, G.A. Vugalter, A.I. Beludanova, Phys. Rev. B 76 (2007) 214401. [39] J.-B. Liu, X.-T. Xie, Y. Wu, Europhys. Lett. 89 (2010) 17006. [40] M.O. Scully, Phys. Rev. Lett. 67 (1991) 1855. [41] M. Fleischhauer, C.H. Keitel, M.O. Scully, C. Su, B.T. Ulrich, S.-Y. Zhu, Phys. Rev. A 46 (1992) 1468. [42] See special issue on slow light: R.W. Boyd, C. Denz, O. Hess, E. Paspalakis (Eds.), J. Opt. 12 (2010) 100301. [43] P.W. Milonni, Fast Light, Slow Light, and Left-Handed Light, Institute of Physics, Bristol, 2005. [44] W.M. Robertson, C. Baker, C. Brad Bennett, Am. J. Phys. 72 (2004) 255. [45] A. Santillán, S.I. Bozhevolnyi, Phys. Rev. B 84 (2011) 064304. [46] W.M. Robertson, J. Pappafotis, P. Flannigan, J. Cathey, B. Cathey, C. Klaus, Appl. Phys. Lett. 90 (2007) 014102. [47] J. Mobley, R.E. Heithaus, Phys. Rev. Lett. 99 (2007) 124301.