Effective modes for general setups - Effective modes for linear Gaussian optomechanics. I. Simp

Chapter VI: Effective modes for linear Gaussian optomechanics. I. Simplify-

6.3 Effective modes for general setups

where we have used that (in the continuum limit)

∫ τ 0

dt|u₁(t)|²= 1 (6.64)

because (as explained in Sec. 6.2.2) the transformation matrix relating ˆaoutand ˆcout

is unitary, so its rows and columns are normalized to 1.

Finally, we note that a similar analysis can be performed to show that the test mass’

quantum state conditioned on n clicks is |ni, and the probability of measuring n clicks is |dn|² = tanh²ⁿr/cosh²r. In addition, the conditional scheme described in Ref. [3] can be extended to setups driven by two tones (instead of just a blue-detuned or red-detuned laser). For instance, consider the setup described in Ref. [7], where a cavity is driven by two tones with frequencies centered around the cavity’s resonant frequency. In the good cavity limit, where the cavity linewidth is much smaller than the mechanical resonant frequency, the authors show that the optomechanical Hamiltonian can be approximated to be squeezing between a squeezed mechanical operator and the driving light. As a result, the analysis performed in Ref. [3] and in this section also applies to the setup considered in Ref. [7]. The only difference is that the mechanical ladder operators would have to be replaced with squeezed ladder operators.

with

w =

˜ aout(t₁)

...

a˜out(tN = τ)

˜ a^†_out(t1)

...

a^†_out(tN = τ) bˆ₁(τ)

...

bˆn(τ) bˆ^†₁(τ)

...

bˆ^†_n(τ) ª

≡ aˆout

bˆ_τ

, (6.66)

whereτis any time after the experiment began, ãoutand ãinis defined in Eq. (6.12), and ˆb1through ˆbnare the ladder operators corresponding to the degrees of freedom of the optomechanical system of interest. âout contains the output optical modes and is anN-size vector (because we’ve discretized time intoN intervals), and ˆb_τ is ann-size vector. vcontains the input optical modes and the system modes evaluated at the initial time of the experimentt1:

v =

˜ ain(t₁)

...

a˜in(τ) a˜_in^† (t1)

...

˜ a_in^† (τ) bˆ1(t1)

...

bˆn(t₁) bˆ^†₁(t₁)

...

bˆ^†n(t1) ª

≡ aˆin

bˆ₀

. (6.67)

M is not an arbitrary matrix because we require that the output light modes be separate degrees of freedom in the same way that the input light modes are. We

must have

MΩM^†= Ω, (6.68)

where the commutation matrixΩkl ≡ h vk,v_l^†i

is given by

Ω = JN 0 0 Jn

. (6.69)

For any positive integerz, Jz is

Jz ≡ Iz 0 0 −Iz

, (6.70)

whereIz is the identity matrix of sizez.

In contrast to Eq. (6.10), we have included both annihilation and creation ladder operators in Eq. (6.65). In general, we cannot, as in Eq. (6.4), apply a rotating wave approximation that would reduce the optomechanical interaction to only the beamsplitter or squeezing interaction. Moreover, we have assumed that the environment consists only of optical modes. Thermal noise can be incorporated by having an explicit model of the thermal bath as a collection of harmonic oscillators. Such models are usually used to derive the Langevin equations of motion [4]. The thermal bath ladder operators can be included in ˆain, and their time-evolved counterparts in

aout. If we include them, then the effective modes we derive in this section would be super-modes consisting of both light and bath ladder operators.

We can decompose Minto four blocks:

M ≡ T H

G B

, (6.71)

where T ∈ M2N×2N relates the output optical fields to the input optical fields, B ∈ M2n×2n relates the system modes at τ to their initial state ˆb₀, H ∈ M2N×2n

specifies the dependence of ˆa_out on ˆb₀, andG ∈ M2n×2N specifies the dependence of ˆb_τ on the input modes ˆa_in. Becausev andwcontain ladder operators and their adjoints,T must have the following structure

T ≡ T₁ T₂ T₂^∗ T₁^∗

∈M2N×2N, (6.72)

whereT₁ ∈ M^N×N indicates how the input optical fields get mixed amongst each

other, andT₂ carries information about their multi-mode squeezing. Similarly, H andGmust have the following structure:

H =

h®₁ ... h®n h˜₁ ... h˜n

, (6.73)

G^† =

g®₁ ... g®n g˜₁ ... g˜n

, (6.74)

where ˜sdenotes the dual of a vector®s

s ≡ KNs®^∗, with

KN ≡ 0 IN

IN 0

(6.75) a ’switching’ matrix that switches the top and bottom halves of®s.

6.3.2 The effective modes that simplifyM’s structure

From Eqs. (6.65) and (6.71), the system modes interact with the neffective input modes Gaˆin. In general, Gaˆin does not form a conjugate set of modes, as their commutation relation is not equal toJn. Consequently, we will choose theneffective input modes to be

Aˆin= S_v^†Gaˆin (6.76)

where ˆA_incontains the annihilation and creation operators ofneffective modes, and S_vensures that

hAˆ_in,Aˆ^†_ini

= Jn, (6.77)

S_v^†GJNG^†S_v = Jn. (6.78) Note that since bottom half of ˆAinhas to be the hermitian conjugate of the top half, S_vhas to be of the form

S_v =

Sv KnS_v^∗

, (6.79)

whereSv is anN ×N matrix.

In practice, we can constructS^†_v with a symplectic Gram-Schmidt procedure. We

pick the first effective input mode to be Aˆin

1 = N_in,1⁻¹g®^†₁aˆin, Aˆin

^†

1 =N_in,1⁻¹g˜₁^†aˆin, (6.80) N_in,1⁻¹ =

g®₁^†JNg®₁

, (6.81)

where we’ve assumed that g®₁^†JNg®₁ > 0. If it isn’t, then ˜g₁^†JNg˜₁ must be positive, andN_in,1⁻¹g˜₁^†aˆin corresponds to an annihilation operator with its conjugate given by N_in,1⁻¹g®₁^†aˆ_in. To construct the second effective mode, we find a vector, ®l₂ that is a linear combination ofg®₁, ˜g₁andg®₂, and that JN-orthogonal tog®₁and ˜g₁:

®l^†JNg®₁=0, ®l^†JNg˜₁=0. (6.82) We then continue this process until we’ve exhausted all the rows ofG.

In Appendix 6.7 (see Eqs. (6.153) and (6.156)), we show that Eq. (6.68) implies that

H^†JNT =

H^†JNH−Jn

B⁻¹G, (6.83)

T JNG^† = −H JnB^†. (6.84)

The first of these equations tells us that theneffective output modes in

Aôut = S_u^†H^†JNaôut (6.85) depend only on the system modes and Â_in = S_v^†Gaˆ_in. Similarly to S_v, S_u is a 2n×2nmatrix that ensures that

hAˆout,Aˆ^†_out

i = Jn. (6.86)

Note thatS_uhas to be of the form Su =

Su −KnS_u^∗

, (6.87)

whereSuis anN×Nmatrix, because the bottom half of Âouthas to be the hermitian conjugate of the top half, andJN introduces a minus sign to the bottom half of âin. If we can show that no other effective output modes than Âout interact with the system modes and Âin, then we’ve realized the desired narrative ofnsystem modes andneffective optical modes interacting with each other and nothing else. Consider

a mode

Oˆ ≡ ®l^†aˆ_out (6.88)

that commutes with ˆAout, then

®l^†J_N²H =0= ®l^†H (6.89) asS_uis invertible. Using Eqs. (6.65) and (6.71), we deduce that Ôdoesn’t depend on the system modes. Furthermore, Ôdepends on the effective input mode®l^†T^†aˆ_in, which commutes with Â_inbecause of Eqs. (6.76) and (6.84).

The system modes and ˆAinand ˆAout interact in the following way:

« Aˆout

bˆ_τ Cˆout

= Mn 0 0 Mot her

« Aˆin

bˆ₀ Cˆin

, (6.90)

where ˆCinand ˆCout are effective modes of the environment that commute with ˆAin

and ˆAout, respectively. Mot her tells us how the ˆCinevolve into the ˆCout, and doesn’t affect the system modes’ dynamics. Mnis

Mn= S_u^†H^†JNT JNG^†S_vJn S_u^†H^†JNH GJNG^†S_vJn B

, (6.91)

where we’ve used Eqs. (6.65) and (6.71), and we’ve used that since S_v^†G in Eq.

(6.76) is a symplectic matrix, we have that

S_v^†G −1

= JNG^†S_vJn. (6.92)

By using Eq. (6.84),we can remove Mn’s dependence onT: Mn = −S_u^†H^†JNH JnB^†S_vJn S_u^†H^†JNH

GJNG^†S_vJn B

. (6.93)

Moreover, we can use Eq. (6.154) to eliminate Mn’s dependence on G, and Eqs.

(6.83) and (6.84) to eliminateMn’s dependence onH. We obtain Mn = −S_u^† Jn−B^†JnB

JnB^†S_vJn S_u^† Jn−B^†JnB Jn−BJnB^†

S_vJn B

. (6.94)

Finally, we note that Appendix 6.7 offers an alternative derivation of the effective modes. We derive the effective modes from the constraints they have to satisfy.

6.3.3 Discussion

The effective modes we have developed in this article simplify the structure of the dynamics, as shown in Fig. 6.1. However, in general they do not simplify the structure of the entanglement between the optomechanical system and its environment.

We argue for this statement with a simple example.

Consider the hypothetical configuration of effective modes shown in Fig. 6.4, where the system degrees of freedom interact only with the effective modes Â⁽¹⁾_in through Â⁽ⁿ⁾_in in a beam-splitter type interaction that swaps the states of the system and effective environment modes. Assume that the initial state of ˆb₁through ˆbnis vacuum, so Â⁽¹⁾_out through Â⁽ⁿ⁾_out will be in vacuum.

bˆ1(τ) through ˆbn(τ) will inherit the state of Â⁽¹⁾_in through Â⁽ⁿ⁾_in, which could be entangled with Â⁽ⁿ_in⁺¹⁾ through Â^(N)_in , because the effective modes are, in general, of the form

Aˆin=

∫ τ 0

dt L(t)aˆin(t)+

∫ τ 0

dtK(t)aˆ^†_in(t), (6.95) whereL(t)andK(t)are arbitrary functions.

As a result, even though the system degrees of freedom do not interact with ˆA⁽ⁿ_in⁺¹⁾ through ˆA^(N)_in , they could still be entangled with them. Our formalism does not, in general, say anything about the entanglement of the system of interest with the environment.

Dalam dokumen Testing alternative theories of Quantum Mechanics with Optomechanics, and effective modes for Gaussian linear (Halaman 162-168)