PHYSICAL REVIEW A 79, 024301 共2009兲
Generating EPR beams in a cavity optomechanical system Zhang-qi Yin1,2 and Y.-J. Han2 1
Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, China FOCUS Center and MCTP, Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA 共Received 4 December 2008; published 3 February 2009兲
2
We propose a scheme to produce continuous variable entanglement between phase-quadrature amplitudes of two light modes in an optomechanical system. For proper driving power and detuning, the entanglement is insensitive with bath temperature and Q of the mechanical oscillator. Under realistic experimental conditions, we find that the entanglement could be very large even at room temperature. DOI: 10.1103/PhysRevA.79.024301
PACS number共s兲: 03.67.Bg, 42.50.Wk, 07.10.Cm
Entanglement is the key resource of the field of quantum information. Light is the perfect medium to distribute entanglement among distant parties. Entangled light with continuous variable 共CV兲 entanglement between phasequadrature amplitudes of two light modes is widely used in teleportation, entanglement swap, dense coding, etc. 关1兴. This type of entangled state is also called the Einstein-PodolskyRosen 共EPR兲 state. The EPR beams have been generated experimentally by a nondegenerate optical parameter amplifier 关2兴, or Kerr nonlinearity in an optical fiber 关3兴. The latter one is simpler and more reliable. The Kerr nonlinearity is used to generate two independent squeezed beams. With interference at a beam splitter, the EPR entanglement is obtained between output beams. However, Kerr nonlinearity in fiber is very weak, which limits entanglement between output beams. It was found that strong Kerr nonlinearity appeared in an optomechanical system consisting of a cavity with a movable boundary 关4–6兴. Besides, the single-mode squeezing could be made insensitive with thermal noise 关5兴, which makes the scheme very attractive. However, the frequency of output squeezed beams cannot be made identical, which makes interference difficult. Then it was generalized to two-mode schemes in order to generate EPR beams without interference 关7–11兴. However, they are either very sensitive to thermal noises 关7,8,10,11兴, or require a ultrahigh mechanical oscillator Q ⬃ 108 to suppress thermal noise effects 关9兴, which is 2 to 3 orders higher than the present available parameters 关12兴. The practical scheme to generate EPR beams in an optomechanical system needs to overcome these problems. In this paper we propose a practical scheme to produce EPR beams in an optomechanical system, which consists of a whispering-gallery mode 共WGM兲 cavity with a movable boundary. We find that, similarly as the single-mode scheme 关5兴, the thermal noise in the two-mode scheme can be greatly suppressed by adiabatically eliminating an oscillator mode. By precisely tuning the laser power and detuning, the oscillation mode is adiabatically eliminated and two output sideband modes are entangled. Unlike the cavity-free scheme 关9兴, our scheme requires modest oscillator Q. Besides, the output light is continuous in our scheme, other than the pulse in Ref. 关9兴. The most attractive feature of our scheme is that the entanglement between output beams is nearly not changed under different bath temperature and Q of the mechanical oscillator. Within the experimentally available parameters 关13,14兴, we find the maximum two-mode squeezing could be higher than 16 dB under room temperature. The entangle1050-2947/2009/79共2兲/024301共4兲
ment of formation 共EOF兲 between two modes is larger than 5 关15兴. Since the coupling efficiency between the cavity and fiber could be larger than 99% in the WGM cavity system 关16兴, we neglect the coupling induced noises in this paper. As shown in Fig. 1, we consider an optomechanical system consisting of a WGM cavity with a movable boundary. There are two cavity modes a and b with the same frequency but opposite momentum. They are coupling with the same mechanical oscillation mode am and driven by four lasers, two from the right-hand side with frequencies L and L⬘, and the other two from the left-hand side with frequencies L out and L⬘. ain and bin are input lights. aout 1 and a2 are output lights. Two lower mirrors have very high probability 共⬎99% 兲 to reflect the driving lasers. So we neglect the reflecting induced noise for aout and bout. The system Hamiltonian is H = H0 + Hd + HI, where 关17–20兴 † H0 = ប p共a†a + b†b兲 + បmam am ,
Hd = ប
冉
冊
冉
冊
⍀⬘ ⍀⬘ ⍀a ⍀b a+ b e−iLt + ប a a + b b e−iL⬘ t + H.c., 2 2 2 2
† 兲. HI = ប共a†b + ab†兲 + បm共a†a + b†b兲共am + am
共1兲
Here a, b, and am are the annihilation operators for the optical and mechanical modes; p and m are their angular frequency. ⍀ j with j = a , b is the driving amplitude and defined as ⍀ j = 2冑P j / បL, where P j is the input laser power and = 1 / ␥ is the photon loss rate into the output modes. is the coupling strength between cavity modes a and b. For the WGM cavity system, it ranges from 100 MHz to 10 GHz
FIG. 1. 共Color online兲 Experimental setup. Cavity modes a and b, which are driven by four lasers, couple to the mechanical mode a m.
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©2009 The American Physical Society
PHYSICAL REVIEW A 79, 024301 共2009兲
BRIEF REPORTS
关19,20兴. The dimensionless parameter = 共 p / m兲共xm / R兲 is used to characterize optomechanical coupling, with xm = 冑ប / mm the zero-point motion of the mechanical resonator mode 关21兴, m its effective mass, and R a cavity radius. In typical WGM cavity systems we find ⬃ 10−4. We define the normal modes a1 = 共a + b兲 / 冑2 and a2 = 共a − b兲 / 冑2. We suppose the conditions that ⍀a − ⍀b = 0 and ⍀a⬘ + ⍀b⬘ = 0 are satisfied. The Hamiltonian can be written as H = ប共 p + 兲a†1a1 + ប共 p − 兲a†2a2 † + ប ma m am + ប
冉
冊 共2兲
where ⍀1 = ⍀a + ⍀b and ⍀2 = ⍀a⬘ − ⍀b⬘. We define the detuning ⌬1 = L − p − and ⌬2 = L⬘ − p + . As shown in Fig. 1, with beam splitters and Faraday rotator, we can get the output mode of a1 and a2. We assume that both cavity and oscillator modes are weakly dissipating at rates ␥ and ␥m, respectively, where ␥m Ⰶ m. We can get quantum Langevin equations 关22兴 † 兲−i a˙ j = i⌬ ja j − ima j共am + am
2
冉
j=1
冊
␥m in am + 冑␥mam , 2
冉
冉
冊
␥ a j + 冑␥ain j , 2
a˙m = − im 兺 共␣*p a p + ␣ pa†p兲 − im +
共7兲
2
m
batically eliminate the am mode. We get am共兲 ⯝ ␦ 共␣*a1 冑␥m in + ␣a†2兲 − i␦ am . Then we have quantum Langevin equations for a1共兲 and a†2共−兲,
␥ − ia1共兲 = − ig⬘a1共兲 − iga†2共− 兲 − a1共兲 + 冑␥ain 1 共兲 2 in 共兲, + 冑˜amam
共4兲
where thermal noise inputs are defined as correlation in† in in† in† 共t兲 , am 共t⬘兲典 = nm␦共t − t⬘兲, 具am 共t兲 , am 共t⬘兲典 functions 具am in in in† in in† in† 具a j 共t兲 , a j 共t⬘兲典 = 具a j 共t兲 , a j 共t⬘兲典 = 具am 共t兲 , am 共t⬘兲典 = 0, in = 具ain j 共t兲 , a j 共t⬘兲典 = 0, with nm the thermal occupancy number of the thermal bath for the oscillator mode. We suppose that cavity modes couple with the vacuum bath. To simplify Eqs. 共3兲 and 共4兲, we apply a shift to normal coordinate, a j → a j + ␣ j, am → am + . ␣ j and  are c numbers, which are chosen to cancel all c number terms in the transformed equations. We find that they should fulfill the following requirements:  ⯝ −共兩␣1兩2 + 兩␣2兩2兲, and i⌬ j␣ j ⍀j + 2i2m␣ j共兩␣1兩2 + 兩␣2兩2兲 − ␥2 ␣ j − i 2 = 0. Because ␥m Ⰶ m, the imaginary part of  can be neglected. In the limit ⌬ j Ⰷ 22wm共兩␣1兩2 + 兩␣2兩2兲, we find ␣ j ⯝ ⍀ j / 冑␥2 + 4⌬2j . In the limit 兩␣兩 Ⰷ 兩具a p典兩, the Langevin equations are linearized as † 兲 + i⌬⬘j − a˙ j = − im␣ j共am + am
␥m in . am + 冑␥mam 2
With proper detuning and input power, we can always tune the cavity mode amplitude ␣1 = ␣2 = ␣. Define the Fourier components of the intracavity field by a共t兲 1 ⬁ −i共t−t0兲 e a共兲d. In the limit ␦ Ⰷ , ␥m, we can adia= 冑 兰−⬁
␥ − ia†2共− 兲 = ig⬘a†2共− 兲 + iga1共兲 − a†2共− 兲 2
共3兲
a˙m = − im 兺 a†j a j − im +
p=1
␥ † a˙2 = − ida2 − im␣2am − a2 + 冑␥ain 2, 2
⍀j ␥ a j − a j共t兲 + 冑␥ain j 2 2
for j = 1,2,
2
␥ a˙1 = − ida1 − im␣1am − a1 + 冑␥ain 1, 2
a˙m = i␦am − im共␣1*a1 + ␣2a†2兲 −
⍀1 −i t ⍀2 −i⬘ t a 1e L + a2e L + H.c. 2 2
† + បm共a†1a1 + a†2a2兲共am + am兲,
In the limit m Ⰷ ␦ , d , ␥ , ␥m, the Langevin equations 共5兲 and 共6兲 can be simplified as
共5兲
冑␥mamin共兲, + 冑␥a†in 2 共− 兲 − ˜
共8兲
2 / ␦, ˜␥m = 共兩␣兩m / ␦兲2␥m, and g⬘ = g + d. In where g = 2兩␣兩2m Eq. 共8兲, we neglect the phase of ␣ because it is not important. a 共兲 ain共兲 in Denote aជ 共兲 = 共 a†1共−兲 兲, aជ in共兲 = 共 ain†1 共−兲 兲, and aជ m 共兲 2
ain共兲
2
= 共 −amin共兲 兲. We get the following matrix equation: m
in , Aaជ 共兲 = 冑␥aជ in共兲 + 冑˜␥maជ m
共9兲
where
A=
冢
− i +
␥ + ig⬘ 2
− ig
ig
␥ − i + − ig⬘ 2
冣
.
in 冑 Using boundary conditions aout j 共兲 = −a j 共兲 + ␥a j共兲 for j = 1 , 2, we can calculate the output field as in in† in aout 1 共兲 = G共兲a1 共兲 − H共兲a2 共− 兲 + I共兲am 共兲,
* in† * in * in aout† 2 共兲 = G共兲 a2 共兲 − H共兲 a1 共− 兲 − I共兲 am 共− 兲,
共10兲
冊
␥m in am + 冑␥mam , 2
2 where G共兲 = 共2 + ␥4 + g2 − g⬘2 − ig⬘␥兲 / ⌬共兲, ˜ m / ⌬共兲 = ig␥ / ⌬共兲, I共兲 = 共−i + ␥2 − ig⬘ + ig兲 ␥␥ = 共−i + ␥2 兲2 + g⬘2 − g2.
冑
共6兲 where j = 1 , 2 and ⌬⬘j = ⌬ j + 22m共兩␣1兩2 + 兩␣2兩2兲. We suppose that ⌬1⬘ ⬍ 0 and ⌬2⬘ ⬎ 0. We define ␦ = 共⌬2⬘ − ⌬1⬘兲 / 2 − m and d = −共⌬1⬘ + ⌬2⬘兲 / 2.
H共兲 and ⌬共兲
Let us define the dimensionless position and momentum out out out† operators of fields Xout j 共兲 = 关a1 共兲 + a1 共−兲兴 and P j 共兲 out out† = 关a j 共兲 − a j 共−兲兴 / i, for j = 1 , 2. We define the correlation matrix of the output field as Vij = 具共i j + ji兲 / 2典, where
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PHYSICAL REVIEW A 79, 024301 共2009兲
BRIEF REPORTS 6
300K 30K 3K
4
10 5 0 −2
0
ω/γ
2
3
2
4 3 2
1
0 −2
α=1000 α=1300 α=1500
6
EOF
Squeeze(dB)
5
EOF
7
20
1
−1.5
−1
−0.5
0
0.5
ω/γ
1
1.5
0 −2
2
−1.5
−1
−0.5
0
ω/γ
0.5
1
1.5
2
FIG. 2. 共Color online兲 EOF for different temperature. 兩␣兩 is 1000, ␦ / 2 = 10 MHz, d = 0.07␥. The maxima squeezing is larger than 16 dB when T = 300 K.
FIG. 3. 共Color online兲 EOF for different cavity mode amplitude ␣. Here we adopt ␦ / 2 = 10 MHz, T = 300 K, ␥ p / 2 = 3.2 MHz, and d = 0.07␥.
out out out = 共Xout 1 , P1 , X2 , P2 兲. We calculate the correlation matrix with Eq. 共10兲. Up to local unitary transformation, the standard form of it is
spectrum width. The embedded figure shows that the maximum squeezing could be larger than 16 dB at room temperature. As shown in Fig. 3, the bigger the cavity mode amplitude ␣, the larger the output entanglement. Because ␣ j ⯝ ⍀ j / 冑␥2 + 4⌬2j , the output entanglement is proportional to driving amplitude. But the peak of entanglement is splitted into two symmetric peaks when the driving is very strong. The splitting distance is proportional to the driving power. Increasing the driving power can decrease the entanglement too. This is because adiabatical elimination condition Ⰶ ␦ are not valid around peaks for very strong driving. So the driving power should be neither too big nor too small. For the specific ␣ and ␦, we find that there is an optimum d which makes entanglement maximum and the entanglement peaks appear near = 0. The optimum d is
VS =
冢
0
n
0
kx
0
n
0 − kx
kx
0
n
0
0 − kx 0
n
冣
,
共11兲
where n = 兵共2 + ␥4 + g2 − g⬘2兲2 + 共g⬘2 + g2兲␥2 + 关共 + g⬘ − g兲2 2 2 2 ˜ m共2nm + 1兲其 / 兩⌬共兲兩2, + ␥4 兴␥␥ kx = 冑V14 + V24 , where 2 ␥ 2 2 2 V14 = −2g␥共w + 4 + g − g⬘ 兲 / 兩⌬共兲兩 , V24 = 兵2g⬘g␥2 + 关共 + g⬘ 2 ˜ m共2nm + 1兲其 / 兩⌬共兲兩2. This is the symmetric − g兲2 + ␥4 兴␥␥ Gaussian state. The EOF for the symmetric Gaussian states is defined as 关15兴 2
do = 冑共22␣2/␦兲2 + ␥2/4 − 共␣兲2/␦ ,
EF = C+共n − kx兲log2关C+共n − kx兲兴 − C−共n − kx兲log2关C−共n − kx兲兴, 共12兲 where C⫾共x兲 = 共x−1/2 ⫾ x1/2兲2 / 4. V describes an entanglement state if and only if n − kx ⬍ 1. Based on the standard form of matrix 共11兲, we also find that 具␦2共X1 + X2兲典 = 具␦2共P1 − P2兲典 = n − kx. We define the two-mode squeezing as S = −10 log10共n − kx兲. We now estimate the bath noise influence in experimentally accessible conditions 关13兴. The cavity resonant frequency p = 2 ⫻ 300 THz. The oscillator frequency m = 2 ⫻ 73.5 MHz. The mechanical Q factor is about 30 000. The cavity and oscillator mode decay rates are ␥ = 2 ⫻ 3.2 MHz and ␥m = m / Q, respectively. The cavity radius is R = 38 m. The dimensionless coupling parameter is ⯝ 10−4. We find if 0 ⬍ d Ⰶ ␥, the thermal noise does not decrease the maximum entanglement with strong enough input power. This is because we adiabatically eliminate the mechanical mode and suppress the effects of thermal noise. As shown in Fig. 2, the temperature change does not change the maximum entanglement with proper driving and detuning. But the higher the temperature, the less the entanglement
corresponding to squeezing So = −10 log10共4d2o / ␥2兲 and entanglement which is obtained from Eq. 共12兲 with n − kx = 4共do / ␥兲2. It is obvious that the higher the input power, the smaller the optimum d. In the mean time, we find that decreasing the mechanical Q factor nearly does not change the entanglement spectrum if d is around its optimum value and the condition m / Q Ⰶ ␦ is fulfilled. Leaving other parameters unchanged, Q could be as low as 300. Considering the difficulty of increasing the mechanical oscillator Q, the above finding makes our scheme more practical. We also test the stability of our scheme. As shown in Fig. 4, the optimum d is around 0.07␥ if ␣ = 1000, ␦ / 2 = 10 MHz. To maintain such high entanglement, we need to precisely control the d down to 0.02␥ ⬃ 2 ⫻ 60 kHz. d is defined as d = −共⌬1⬘ + ⌬2⬘兲 / 2 = −共⌬1 + ⌬2兲 / 2 − 42m兩␣兩2. The higher the entanglement is needed, the more precise detuning and driving power is required at the same time. To maintain the entanglement as high as Fig. 4, the laser spectrum width should be less than 60 kHz and the driving power fluctuation should be less than 1%. The lower entanglement between two beams is needed to maintain, the larger the optimum d
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PHYSICAL REVIEW A 79, 024301 共2009兲
BRIEF REPORTS 6 d=0.07γ d=0.09γ d=0.05γ
5
EOF
4
3
2
1
0 −2
−1.5
−1
−0.5
0
ω/γ
0.5
1
1.5
2
FIG. 4. 共Color online兲 EOF for different d. Here we adopt m / 2 = 73.5 MHz, T = 300 K, ␥m = m / 30 000, ␥ / 2 = 3.2 MHz, and 兩␣兩 = 1000.
is. Therefore, higher fluctuations of detuning and driving power are allowed. Before concluding, we briefly discuss the approximations we used. Our scheme needs the steady states existing, which requires 具a†j a j典 Ⰶ 兩␣兩2. During numerical calculation, 具a†a典 is in the order of 103, which is much less than 兩␣兩2 ⬃ 106. The other two approximations are the rotating wave approximation m Ⰷ ␦ , d , ␥ , ␥m and adiabatical elimination ␦ Ⰷ , ␥m, which can be fulfilled independently. For ␣ ⬃ 103, the driving
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amplitude ⍀ is in the order of 1011 Hz, which is much lower than the distance between adjacent cavity modes ⌬ = c / 共Rn0兲 ⬃ 5 ⫻ 1012 Hz, where c is the light speed in a vacuum and n0 is the refractive index of silica. Therefore the approximation that one laser only drives one cavity mode is valid. Laser power is needed in the order of 10 mW, which is available in the laboratory. In conclusion, we have proposed a scheme to generate EPR lights in an optomechanical system. Two sideband modes, which couple with the mechanical mode, are driven by lasers. After adiabatically eliminating the mechanical mode, we find that the output sideband modes are highly entangled. The higher the power of the driving laser, the larger entanglement of the output light. To maintain the entanglement, we need to precisely control the driving power and laser frequency at the same time. With proper parameters, the entanglement is insensitive to the thermal noise and mechanical Q factor. We test the scheme by experimentally available parameters. Though in this paper we focus on WGM cavity systems, our scheme can be realized in other optomechanical systems, as long as the mechanical mode frequency is much larger than the cavity decay rate. We thank Lu-ming Duan for helpful discussions. We thank Yun-feng Xiao and Qing Ai for valuable comments on the paper. Z.Y. was supported by the Government of China through CSC 共Contract No. 2007102530兲.
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