Using reservoir-engineering to convert a coherent signal in optomechanics with small optomechanical cooperativity

JID:PLA

AID:24404 /SCO Doctopic: Optical physics

[m5G; v1.208; Prn:17/03/2017; 12:57] P.1 (1-5)

Physics Letters A ••• (••••) •••–•••

1

Contents lists available at ScienceDirect

2

67 68

3

Physics Letters A

4 5

69 70 71

6

72

www.elsevier.com/locate/pla

7

73

8

74

9

75

10

76

11 12 13

Using reservoir-engineering to convert a coherent signal in optomechanics with small optomechanical cooperativity

14 15 16 17 18 19 20

Tao Wang

c

b

, Tie Wang , Changbao Fu , Xuemei Su

a

25 26 27 28

82

a

Key Lab of Coherent Light, Atomic and Molecular Spectroscopy, Ministry of Education, and College of Physics, Jilin University, Changchun 130012, People’s Republic of China b College of Physics, Tonghua Normal University, Tonghua 134000, People’s Republic of China c Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, People’s Republic of China

31 32 33

a r t i c l e

i n f o

Article history: Received 2 January 2017 Received in revised form 13 March 2017 Accepted 14 March 2017 Available online xxxx Communicated by V.A. Markel Keywords: Optomechanical dark mode State conversion Reservoir-engineering

Optomechanical dark mode plays a central role in effective mechanically-mediated conversion of two different cavity fields. In this paper, we present a more efficient method to utilize the dark mode to transfer a coherent signal. When an auxiliary cavity mode is exploited, two approaches are proposed to effectively eliminate the optomechanical bright mode, and only the optomechanical dark mode is left to facilitate state transfer. Even with small cooperativity and different losses for the two target modes, the internal cavity mode-conversion efficiency can also reach unity. © 2017 Elsevier B.V. All rights reserved.

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

65 66

91 92 93 94 95 96 98 99 101 102

1. Introduction

103

In quantum optomechanics, the mechanical state can be prepared and controlled by the cavity fields via radiation pressure [1]. Lots of remarkable progresses have been made, including cooling the mechanical resonator into the ground state [2–4], optomechanically induced transparency [5,6], and the squeezing of the mechanical motion [7–9]. In these progresses, thermal mechanical dissipation is one of the major obstacles. A direct approach to overcome the thermal noise is preparing the mechanical oscillator into the ground state [2–4]. However this approach is not the unique method for some quantum tasks, especially for quantum state conversion between two cavity fields with different frequencies. An indirect approach, exploiting the use of optomechanical dark mode, was proposed in Refs. [10–12] and experimentally studied in Refs. [13–21]. Two recent useful reviews on this topic can be found in Refs. [22,23]. For quantum state transfer, two target cavity modes are simultaneously coupled to a common mechanical resonator, and can be hybridized to two normal modes: an optomechanical bright mode (OBM) and an optomechanical dark mode (ODM). ODM is similar to the idea of dark state in coherent population trapping and electromagnetically induced transparency in the -style atoms [24,25]. ODM is dynamically decoupled from the thermal mechanical motion, and offers an efficient channel for quantum state transfer (see

62 64

90

100

61 63

86

97

37 39

85

89

35

38

84

88

a b s t r a c t

34 36

83

87

29 30

79 81

23 24

78 80

a, b

21 22

77

E-mail addresses: [email protected] (T. Wang), [email protected] (X. Su). http://dx.doi.org/10.1016/j.physleta.2017.03.017 0375-9601/© 2017 Elsevier B.V. All rights reserved.

104 105 106 107 108 109 110 111 112 113 114

Fig. 1. A schematic for signal transfer. The signal is transferred via an optomechanical dark mode (ODM) and an optomechanical bright mode (OBM). The ODM is decoupled from the thermal mechanical noise, so it favors the signal conversion. The OBM suffers from the thermal dissipation. In our paper, we use the idea of reservoir-engineering to control the OBM.

115 116 117 118 119

Fig. 1). The key factor is to reduce the influence of the OBM, which suffers from the thermal noise. In the existing scheme [10,12], the OBM can be suppressed by the mechanism of optomechanically induced transparency [5,6]. The performance of an optomechanical system can be greatly increased by engineering a specific reservoir. A simple method is to couple the mechanical resonator with another auxiliary cavity mode. The first proposal based on this idea is to generate strong entanglement between the two target cavity modes [26], in which the auxiliary mode driven at red-detuned sideband can

120 121 122 123 124 125 126 127 128 129 130 131 132

JID:PLA

AID:24404 /SCO Doctopic: Optical physics

[m5G; v1.208; Prn:17/03/2017; 12:57] P.2 (1-5)

T. Wang et al. / Physics Letters A ••• (••••) •••–•••

2

ity modes and the mechanical resonator can be greatly enhanced [22,23]. The Hamiltonian for this system is

1 2 3 4 6



9 11

†

13 14 15 16 17 18 19

Fig. 2. A schematic for signal transfer in our scheme.

21

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

greatly enhance the dissipation of the OBM (cooling). In addition, quantum-limited amplification with this setup was also discussed in Refs. [27,28]. Recently we found that prominent entanglement between the two target modes can be available under roomtemperature [29]. We offer two methods to control the OBM, which are different from the approach used in Refs. [26–29]. The first approach is to drive the auxiliary mode at blue-detuned sideband. This kind of reservoir-engineering is exploited for the first time in this paper. The second approach is to use the auxiliary cavity mode to parametrically modulate the spring constant of the mechanical resonator at twice its natural frequency. This parametric modulation can induce the squeezing of the oscillator [30,31]. Recently this interaction was exploited to amplify the single-photon nonlinearity [32] and to realize phase-sensitive amplification and squeezing of an optical signal [33]. In our paper, we investigate the coherent signal conversion following the way in Ref. [14]. In section 2, we remove the OBM with blue-sideband detuned driving. In section 3, we eliminate the OBM with parametric modulation. Although the two methods are different, the ultimate results are the same. The optomechanical cooperativity of the two cavity modes is equal to the ideal state conversion, but they can be very small. The losses of the two target modes can be very different. The conclusion is given in section 4.

48

2. Driving at blue-detuned sideband

49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

(2)

The OBM can be controlled completely via introducing a new control parameter. An auxiliary cavity mode is introduced, which is also coupled to the mechanical oscillator. The critical idea in this setup is to control signal conversion by use of the auxiliary cavity mode via the mechanical resonator. This three-terminal four-mode optomechanical setup has been recently discussed in Refs. [26,27,29] for strong entanglement and quantum-limited amplification. Fig. 2 is the schematic for signal transfer in our scheme. The optomechanical system consists of three cavity modes 1, 2, 3 and one mechanical resonator. The three cavity fields are simultaneously coupled to the same mechanical mode. The cavity modes 1, 2 are exploited to transfer a coherent signal. The cavity mode 3 is used to control the OBM. The signal enters into cavity 1 and comes out from cavity 2. The cavity modes 1, 2 are respectively driven by strong driving fields 1, 2 at red-detuned sideband, while the cavity mode 3 is driven by a strong driving field 3 at bluedetuned sideband. The coupling strengths between the three cav-



α˙ 1 = −

κ1

α˙ 2 = −

κ2

β˙ = −

γm

∗

κ3

2 2

2

2

α1 − iG 1 β + κ α1,in e e 1

−i δ t

(3)

aD = If

(4)

κ

+κ

− iG β +

(5)

α˙ D = −

2 2G2

G

+

G2 G



αB − e 1

G 1 G 2 (κ1 − κ2 ) 2G 2

κ α1,in e

κ1 G 22 + κ2 G 21 2G 2

85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105

112 113 114 115 116 117 119

.

2G 2  G1

84

118

,

α B = a B e−iωm t , α D = a D e−iωm t , we have

α˙ B = −

83

111

α3 − iG 3 β,

2 1G1

82

110

∗

G

81

109

β − iG 1 α1 − iG 2 α2 + iG 3 α3 ,

G 2 a1 − G 1 a2

80

108

∗

G

79

107

α2 − iG 2 β,

G 1 a1 + G 2 a2

78

106

,

here γm is the damping rate of the mechanical resonator. The stability conditions for our system are given in Appendix A. The OBM and the ODM can be defined as

aB =

76 77

†

Here H 0 describes the energies of the two cavity modes 1, 2 and the mechanical resonator, their interactions, and the driving interaction by the signal incident on the cavity mode 1. The auxiliary Hamiltonian H a1 describes the energy of the cavity mode 3, the interaction between cavity mode 3 and the mechanical resonator. h¯ is the reduced Planck constant. a1 , a2 , a3 and b are respectively the annihilation operators for the cavity fields 1, 2, 3 and the mechanical oscillator. ωm is the mechanical frequency. If ωc ,i and ωl,i (i = 1, 2, 3) are respectively the frequencies for the three cavity fields and the three driving fields on the three cavities, we have ωm = ωc,1 − ωl,1 = ωc,2 − ωl,2 = ωl,3 − ωc,3 . G 1 , G 2 and G 3 are the effective optomechanical coupling rates between the three cavity modes 1, 2, 3 and the mechanical resonator. a1,in is the annihilation operator for the incident signal. If ω p is the frequency of the signal,  = ω p − ωl,1 is the detuning between the signal and the driving field 1. κ1e is the effective output coupling rate of the cavity modes 1. It should be noticed that we work in the resolvedsideband regime κi  ωm (i = 1, 2, 3) for any cavity mode, and the κi (i = 1, 2, 3) are respectively the total decay rates of the cavity modes 1, 2 and 3. For coherent signal, whose amplitude is much larger than the vacuum fluctuation, we can treat them as classical variables [14]. The noise for our system is briefly discussed in Appendix B. † We define α1 = a1 e −i ωm t , α2 = a2 e −i ωm t , α3∗ = a3 e −i ωm t , β = be −i ωm t and α1,in = a1,in , then the equations of motion for the Hamiltonian can be described as follows

α˙ 3 = −

47

74 75

†

H a1 = −h¯ ωm a3 a3 + h¯ G 3 (a3 b† + a3 b).

12

25

†

71 73

+ b † a2 )

+ ih¯ κ1e (e −i t a1 a1,in − e i t a1,in a1 ),

10

24

69

72

ω

8

23

(1)

† † H 0 = h¯ m (a1 a1 + a2 a2 + b† b) † † + h¯ G 1 (a1 b + b† a1 ) + h¯ G 2 (a2 b

7

22

68 70

H 1 = H 0 + H a1 ,

5

20

67

αD −

κ1e α1,in e−iδt ,

−i δ t

120 121 122 123 124

αD

125 126 127

,

128

G 1 G 2 (κ1 − κ2 ) 2G 2

αB

129 130 131 132

JID:PLA

AID:24404 /SCO Doctopic: Optical physics

[m5G; v1.208; Prn:17/03/2017; 12:57] P.3 (1-5)

T. Wang et al. / Physics Letters A ••• (••••) •••–•••

1

β˙ = −

γm

α˙ 3∗ = −

κ3

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

2

2

26 27 28 29 30 31

34 35

G1

α (B1−) =

38 39 40 41 42

G A + D1

α (D1−) = (

G2 G

−

45 46 47 48 49 50 51 52 53 54 55 56 57

60 61 62 63 64

κ

e 1

− iδ

2

B

G A + D1

G 21

G 21

+

G 22 ,

α1,in , 

) κ1 2

κ1e

− iδ

D1 = (

2

α1,in ,

(7)

G 22

G 23

− i δ) − κ3 . − iδ 2

(8)

= 0) =

1 − C3

G1

G 1 + C1 + C2 − C3

√ 2 η1 √ α1,in ,

κ1

(9)

α (D1−) (δ

= 0) =

√ κ1 G 2 1 + κ2 C 1 + C 2 − C 3 2 η1 G

1 + C1 + C2 − C3

√

κ1

α1,in .

(10)

4G 2

κ1e κ1

of the three cavity fields. We define η1 = for the output cou1. Using the input–output relation, pling ratios of the cavity mode  the outgoing signal is α2,out = κ2e α2− . κ2e is the effective output coupling rate of the cavity modes 2. It is clear that, with assistance of reservoir-engineering, the OBM in Eq. (9) can be zero if the following condition is satisfied

C 3 = 1.

α D − (δ = 0) =

γm 2

77 78 79 80

Fig. 3. The conversion efficiency in our pure ODM scheme (blue-solid ling) and in the existing scheme (red-dashed line) as a function of C 1 , and C 2 = C 1 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

81 82 83 84 85

2G 2

If C 3 = 1, α B = Gi ( 2m − κ 3 )β = 0 can be obtained directly. Thus 3 in our scheme, the introduction of the auxiliary mode can directly prohibit the excitation of the OBM, which is very different from the mechanism of optomechanically induced transparency [10,12]. G α −G α For α2− = 2 B − G 1 D − , the cavity mode-conversion efficiency at δ = 0 can be given by

(14)

86 87 88 89 90 91 92 93 94 95

It is clear that the conversion efficiency is immune to the mechanical dissipation. Fig. 3 compares the conversion efficiency in our pure ODM scheme (blue-solid line) with that in the existing scheme (red-dashed line). In the existing scheme [10–23], the 1 C2 conversion efficiency is χ (0) = η1 η2 (1+4C . When C 1 = C 2 and C +C )2 1

2

η1 η2 = 1, the conversion efficiency can approach unity. If the con-

version efficiency reaches 0.99, C 1 = C 2 > 99 is needed [20]. In our scheme, the C 1 and C 2 do not need to be much larger than unity, so G 1 and G 2 can be very small.

96 97 98 99 100 101 102 103 104 106 107

We also notice that parametric modulation of the mechanical resonator can be also used to remove the OBM. Ref. [33] pointed out that the unusual dynamics in optomechanics with mechanical parametric driving has interesting results for optomechanically induced transparency, and the cavity spectra function can be negative. Here we focus on the point that the cavity spectra function is zero. The auxiliary mode 3 can be used to parametrically modulating the spring constant of the mechanical oscillator at twice the mechanical frequency [32], the effective Hamiltonian for this setup is

G

C1 + C2

√

κ1

α1,in ,

β − iG α B + iG 3 α3∗ = 0,

α3∗ − iG 3 β = 0.

(15)

λ

(12)

(13)

108 109 110 111 112 113 114 115 116 117 118 119

H 2 = H 0 + H a2 ,

so it is in fact independent of the thermal noise even the losses of the two cavity modes 1 and 2 are not equal. We can provide a simple view for this result. When δ = 0, the last two equations in Eq. (6) are

−

76

(11)

Then the ODM can be further reduced as √ κ1 G 2 κ2 C 1 + C 2 2 η1 (1 )

2

75

3. Parametric modulation

Here C i = γ κi (i = 1, 2, 3) are the optomechanical cooperativity m i

κ3

74

105

and

−

73

|α2,out |2 4C 1 C 2 χ (1 ) = = η1 η 2 . |α1,in |2 (C 1 + C 2 )2

Here (1) denotes the ODM removing scheme with blue-detuned sideband driving. For δ = 0, Eq. (7) can be reduced as

α (B1−) (δ

72

γ

G1G2 G1G2 − κ2 , B = κ1 − i δ − iδ 2 2

γm

71

so it suf-

+ κ2 , A = κ1 − iδ − iδ 2 2

65 66

70

where

58 59

κ1

G1

43 44

69

(6)



D1

36 37

68

α3∗ − iG 3 β.

fers from the thermal noise. However, the magnitude of the OBM is related to the amplitude of the mirror motion, which are both suppressed due to optomechanically induced transparency when G  0 is satisfied. Let α1 = α1− e −i δt , α2 = α2− e −i δt , α B = α B − e −i δt , α D = α D − e−iδt , α3∗ = α3∗+ e−iδt and β = β− e−iδt , then the steady state solution can be derived. Thus we have

32 33

67

oscillator with an effective coupling rate G =

24 25

β − iG α B + iG 3 α3∗ ,

In the existing scheme [10,12], G 3 = 0 and κ1 = κ2 . The ODM is absolutely decoupled from the mechanical resonator, so it is immune to the thermal noise. The OBM is coupled  to the mechanical

22 23

3

H a2 = − (e −2i ωm t (b† )2 + e 2i ωm t (b)2 ). 2

121

(16)



α˙ 1 = − α˙ 2 = −

κ2

β˙ = −

γm

2 2

2

124 125 126 127

α1 − iG 1 β + κ1e α1,in e−iδt ,

128 129

α2 − iG 2 β, β − iG 1 α1 − iG 2 α2 + i λβ ∗ .

122 123

H a2 describes the parametric modulation interaction of the mechanical oscillator. λ is the mechanical parametric driving strength. The equations of motion for H 2 are

κ1

120

130 131

(17)

132

JID:PLA

AID:24404 /SCO Doctopic: Optical physics

[m5G; v1.208; Prn:17/03/2017; 12:57] P.4 (1-5)

T. Wang et al. / Physics Letters A ••• (••••) •••–•••

4

1

Thus the expressions for the OBM and the ODM can be derived as

2 3 4

α (B2−)

=

7 8 9 10 11

α (D2−)

κ1e

α1,in , G A + D 2 κ21 − i δ

5 6

D2

G1



=(

G2 G

κ

B

G1

−



) α1,in , G A + D 2 κ21 − i δ

D2 = (

γm 2

− i δ) −

|λ|2 ( γ2m − i δ) +

13 15 16

19 20 21

α (B2−) (δ = 0) =

G 1 − t + C1 + C2

26 27 28 29 30 31 32 33 34

37 38 39 40 41 42

.

(19)

√ 2 η1 √ α1,in ,

κ1

G

1 − t + C1 + C2

√

κ1

α1,in ,

t=

(21)

47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

71 72 73 74 75 76 77 78 79 80 81

85 86 87

Appendix A. Stability conditions

88

γm2 (1 + C 1 + C 2 )

(22)

.

It is clear that Eq. (20) can be set to zero if 1 − t = 0 or λ = 1 + C 1 + C 2 is satisfied. We can see that the ODM is simplified

as

√

κ1 G 2 κ2 C 1 + C 2 2 η1 = 0) = √ α1,in , G C1 + C2 κ1

(23)

and the cavity mode-conversion efficiency at δ = 0 can be given by

χ ( 2 ) = η1 η 2

4C 1 C 2

. (C 1 + C 2 )2

The stability conditions of the three-terminal four-mode optomechanical setup in the first approach with blue-detuned driving have been discussed in Refs. [26,27,29]. The effective dissipaγ

⎛ ⎜

A=⎜ ⎝

(24)

These results for the ODM and the conversion efficiency with parametric modulation are the same as that with the blue-detuned driving. For the OBM is removed, the ODM is decoupled from the mechanical resonator. Other approaches to remove the OBM will offer the same results.

γ

2G 23 κ3

2G 22 κ2

γ

− κ21

⎞

− κ22 −iG 2

0 −iG 1 0

0

91 92 93 94 95 96 97 98

−iG 1 0 −iG 2 0 ⎟ ⎟. γm −2 iG 3 ⎠ κ3 −iG 3 − 2

0

90

99

(25)

100 101 102

According to the Routh–Hurwitz criterion in Ref. [43], we have

103

p 0 = 1,

105

p1 = p2 =

4. Conclusion In this paper, we point out that the OBM can be eliminated for state conversion between two cavity fields, and only the ODM is left for the task. Two approaches are put forward to elaborate this idea. Although the methods are different, the results obtained are the same. For the best conversion efficiency, it only needs that the two optomechanical cooperativity are equal, that C 1 = C 2 , except for the OBM removing condition. These results are important for designing new optomechanical interfaces for hybrid quantum networks, especially can be used for the weak light control. For the coupling strength can be very small, our result pave the way for quantum state transfer in the ultra-weak coupling regime. In fact, we offer a new perspective to utilize the dark state or dark mode, which plays an important role in many quantum tasks, such as coherent population trapping and electromagnetically induced transparency [24,25]. Optomechanical systems can be coupled to many other quantum systems except for various cavity modes, such as Bose–Einstein condensate [34], superconducting qubits [35–37] and nitrogen-vacancy centers in diamond [38,39]. Thus our method can be also used to investigate the qubit-state transfer between such quantum systems via the mechanical oscillator [40,41]. In addition, similar ideas that the bright mode or

2G 21 κ1

+ − = 2m (1 + C 1 + tion of the resonator e2f f = 2m + C 2 − C 3 ) > 0 is needed for stability. In our scheme, C 3 = 1 and C 1 = C 2 > 0 can satisfy this condition. A detailed result for the stability of this special configuration is useful, so we provide it here. From Eq. (4), the condition matrix is

κ1

p3 =

2

κ1 κ2

+

4 G 21

+

κ1 κ2 κ3 8

+( p4 =

κ2

+

2

+

45 46

70

84

This research is supported by National Natural Science Foundation of China, Grant No. 11174109 and Grant No. 11404242.

43 44

69

89

4|λ|2

α (D2−) (δ

68

83

(20)

γm √ 2

67

82

Acknowledgement

where

35 36

+

G 22 κ2 −i δ 2

√ κ1 G 2 1 − t + κ2 C 1 + C 2 2 η1

α (D2−) (δ = 0) =

24 25

1−t

G1

22 23

G 21 κ1 −i δ 2

Here (2) denotes the ODM removing scheme with parametric modulation. When δ = 0, Eq. (18) can be reduced as

17 18

(18)

where

12 14

e 1

bright state can be removed can be also used in electromagnetically induced transparency with various quantum systems, such as atoms, trapped ions and superconducting circuits. For the controlling strength can be very weak, the signal propagation in this optomechanical transducer can be very slow. The system can be exploited to use as a quantum buffer and a quantum memory. Many works based on this paper should be further clarified. A critical problem is that whether we can use weak fields to control a strong signal. In addition, for nonclassical quantum state conversion and single-photon-level quantum state transfer, a fullquantum treatment should be further discussed [42]. The added noises result from the thermal noise and the squeezing noise (see Appendix B), which should be further reduced. The transfer fidelity in diverse approaches to remove the OBM should be investigated.

κ1 2

κ3

+

2

κ1 κ3

+

4 G 22

+

+

16

−

2

107

+

4

κ1 γm κ2 γm 4

4

+

κ3 γm 4

108

+

109 110

G 23 , 8

)G 22

111

+

+(

κ1 κ3

+

2

106

,

κ2 κ3

κ1 κ2 γm

κ3

κ1 κ2 κ3 γm

+

γm

4

104

κ1 κ3 γm

κ2 2

8

+

G 22 +

κ3 2

)G 21

κ2 κ3 4

+

κ2 κ3 γm −(

G 21 −

κ1 2

+

κ1 κ2 4

112

+

8

κ2 2

113 114

)G 23 ,

115

G 23 ,

(26)

116 117

thus

118

T 0 = p0,

120

119 121

T 1 = p1 ,

122

T 2 = p1 p2 − p0 p3 ,

123 124

T 3 = p 3 T 2 − p 21 p 4

125

T 4 = p4 T 3 .

(27)

127

The expressions for T 0 –T 4 are very tedious, and they are not explicitly written here. The stability conditions are T 0 > 0, TT 1 > 0, T2 T1

> 0,

T3 T2

> 0,

126

T4 T3

0

= p 4 > 0. For C 3 = 1, we can verify that these

conditions are all satisfied, so the optomechanical system considered in our paper is stable.

128 129 130 131 132

JID:PLA

AID:24404 /SCO Doctopic: Optical physics

[m5G; v1.208; Prn:17/03/2017; 12:57] P.5 (1-5)

T. Wang et al. / Physics Letters A ••• (••••) •••–•••

1

Appendix B. Noise

2 3 4 5 6 7 8 9

In our paper, we transfer a coherent signal with large amplitude, which is much larger than the vacuum fluctuation, so the noise can be omitted. However if we treat a very weak quantum signal, the noises in this system are needed to consider. Here we give a brief discussion for it, detailed results will be given in future paper. The noises for the three cavity modes and the resonator are δa1 , δa2 , δa3 , δb, and they satisfy the following equations

10 11

δa˙ 1 = −i ωm δa1 −

κ1

δa˙ 2 = −i ωm δa2 −

κ2

12 13 14 15 16

δb˙ = −i ωm b − † + iG 3 δa3

17 18 19

†

†

21 22 23 24 25 26

δ S out ,2 (ω) = η2

29 30

33 34 35 36 37

+

√

κ1 δa1,in , κ2 δa2,in ,

b + iG 1 δa1 − iG 2 δa2

√

γm δbin , κ3 † 2

δa3 − iG 3 b +

γm γ2 (ω + ωm

+ η2 η 3

28

32

2

2

δa2 − iG 2 b +

√

√

κ3 δa†3,in .

(28)

Here δa1,in , δa2,in , δa3,in , δb in are the input noises for the three cavity modes and the oscillator. With the simplification method used in [13] (see the supplementary information), the output noises spectrum is

27

31

γm

δa˙ 3 = −i ωm δa3 −

20

2

δa1 − iG 1 b +

2G 2

)2

+ (γe f f

nb /2)2

γ2 γ3

(29)

, (ω + ωm )2 + (γe f f /2)2 2G 23 κ3

here γ2 = κ 2 , γ3 = and nb is the number of mechanical 2 quanta due to thermal fluctuations. We assume that the three cavity modes are all nearly in the vacuum state and the quantum back-action noise can be neglected. The second noise is introduced by the auxiliary cavity mode with squeezing. When C 3 = 1, γ η γ the added noise for the idea transfer is nadd = η1 γm nb + η3 γ3 1

2G 21 κ1

38

(γ1 =

39

can be omitted, we need |α1,in |2  nadd .

1

1

1

), which is finite under our conditions. If the added noise

40 41

References

42 43 44 45 46 47 48 49

[1] M. Aspelmeyer, T.J. Kippenberg, F. Marquardt, Rev. Mod. Phys. 86 (2014) 1391. [2] A.D. O’Connell, M. Hofheinz, M. Ansmann, R.C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J.M. Martinis, A.N. Cleland, Nature (London) 464 (2010) 697. [3] J. Chan, T.P. Mayer Alegre, A.H. Safavi-Naeini, J.T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, O. Painter, Nature (London) 478 (2011) 89. [4] J.D. Teufel, T. Donner, D. Li, J.W. Harlow, M.S. Allman, K. Cicak, A.J. Sirois, J.D. Whittaker, K.W. Lehnert, R.W. Simmonds, Nature (London) 475 (2011) 359.

5

[5] S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, T.J. Kippenberg, Science 330 (2010) 1520. [6] A.H. Safavi-Naeini, T.P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J.T. Hill, D.E. Chang, O. Painter, Nature (London) 472 (2011) 69. [7] E.E. Wollman, C.U. Lei, A.J. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A.A. Clerk, K.C. Schwab, Science 349 (2015) 952. [8] J.M. Pirkkalainen, E. Damskägg, M. Brandt, F. Massel, M.A. Sillanpää, Phys. Rev. Lett. 115 (2015) 243601. [9] F. Lecocq, J.B. Clark, R.W. Simmonds, J. Aumentado, J.D. Teufel, Phys. Rev. X 5 (2015) 041037. [10] Y.D. Wang, A.A. Clerk, Phys. Rev. Lett. 108 (2012) 153603. [11] L. Tian, Phys. Rev. Lett. 108 (2012) 153604. [12] Y.D. Wang, A.A. Clerk, New J. Phys. 14 (2012) 105010. [13] J.T. Hill, A.H. Safavi-Naeini, J. Chan, O. Painter, Nat. Commun. 3 (2012) 1196. [14] C.H. Dong, V. Fiore, M.C. Kuzyk, H.L. Wang, Science 338 (2012) 1609. [15] Y. Liu, M. Davanco, V. Aksyuk, K. Srinivasan, Phys. Rev. Lett. 110 (2013) 223603. [16] J. Bochmann, A. Vainsencher, D.D. Awschalom, A.N. Cleland, Nat. Phys. 9 (2013) 712. [17] T. Bagci, A. Simonsen, S. Schmid, L.G. Villanueva, E. Zeuthen, J. Appel, J.M. Taylor, A. Sørensen, K. Usami, A. Schliesser, E.S. Polzik, Nature (London) 507 (2014) 81. [18] R.W. Andrews, R.W. Peterson, T.P. Purdy, K. Cicak, R.W. Simmonds, C.A. Regal, K.W. Lehnert, Nat. Phys. 10 (2014) 321. [19] C.H. Dong, V. Fiore, M.C. Kuzyk, L. Tian, H.L. Wang, Ann. Phys. 527 (2015) 100. [20] F. Lecocq, J.B. Clark, R.W. Simmonds, J. Aumentado, J.D. Teufel, Phys. Rev. Lett. 116 (2016) 043601. [21] C.F. Ockeloen-Korppi, E. Damskägg, J.M. Pirkkalainen, T.T. Heikkilä, F. Massel, M.A. Sillanpää, Phys. Rev. X 6 (2016) 041024. [22] L. Tian, Ann. Phys. (Berlin) 527 (2015) 1. [23] C.H. Dong, Y.D. Wang, H.L. Wang, Nat. Sci. Rev. 2 (2015) 510. [24] E. Arimondo, Prog. Opt. 35 (1996) 257. [25] M. Fleischhauer, A. Imamoglu, J. Marangos, Rev. Mod. Phys. 77 (2005) 633. [26] Y.D. Wang, A.A. Clerk, Phys. Rev. Lett. 110 (2013) 253601. [27] A. Metelmann, A.A. Clerk, Phys. Rev. Lett. 112 (2014) 133904. [28] A. Nunnenkamp, V. Sudhir, A.K. Feofanov, A. Roulet, T.J. Kippenberg, Phys. Rev. Lett. 113 (2014) 023604. [29] T. Wang, R. Zhang, X.M. Su, Commun. Theor. Phys. 65 (2016) 596. [30] A. Szorkovszky, G.A. Brawley, A.C. Doherty, W.P. Bowen, Phys. Rev. Lett. 110 (2013) 184301. [31] A. Pontin, M. Bonaldi, A. Borrielli, F.S. Cataliotti, F. Marino, G.A. Prodi, E. Serra, F. Marin, Phys. Rev. Lett. 112 (2014) 023601. [32] M.A. Lemonde, N. Didier, A.A. Clerk, Nat. Commun. 7 (2016) 11338. [33] B.A. Levitan, A. Metelmann, A.A. Clerk, New J. Phys. 18 (2016) 093014. [34] S. Singh, H. Jing, E.M. Wright, P. Meystre, Phys. Rev. A 86 (2012) 021801. [35] C.P. Sun, L.F. Wei, Y.X. Liu, F. Nori, Phys. Rev. A 73 (2006) 022318. [36] T.T. Heikkilä, F. Massel, J. Tuorila, R. Khan, M.A. Sillanpää, Phys. Rev. Lett. 112 (2014) 203603. [37] J.M. Pirkkalainen, S.U. Cho, F. Massel, J. Tuorila, T.T. Heikkilä, P.J. Hakonen, M.A. Sillanpää, Nat. Commun. 6 (2015) 6981. [38] O. Arcizet, V. Jacques, A. Siria, P. Poncharal, P. Vincent, S. Seidelin, Nat. Phys. 7 (2011) 879. [39] D.A. Golter, T. Oo, M. Amezcua, K.A. Stewart, H.L. Wang, Phys. Rev. Lett. 116 (2016) 143602. [40] P. Rabl, S.J. Kolkowitz, F.H.L. Koppens, J.G.E. Harris, P. Zoller, M.D. Lukin, Nat. Phys. 6 (2010) 602. [41] J. Zhou, Y. Hu, Z.Q. Yin, Z.D. Wang, S.L. Zhu, Z.Y. Xue, Sci. Rep. 4 (2014) 6237. [42] A.H. Safavi-Naeini, O. Painter, New J. Phys. 13 (2011) 013017. [43] E.X. DeJesus, C. Kaufman, Phys. Rev. A 35 (1987) 5288.

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115

50

116

51

117

52

118

53

119

54

120

55

121

56

122

57

123

58

124

59

125

60

126

61

127

62

128

63

129

64

130

65

131

66

132