Measurement-induced macroscopic superposition
states in cavity optomechanics

Measurement-induced macroscopic superposition
states in cavity optomechanics

Jonas S. Neergaard-Nielsen, Ulrich B. Hoff,

Johann Kollath-Bönig, Ulrik L. Andersen

QPIT, Dept. of Physics, Technical University of Denmark

Jonas S. Neergaard-Nielsen, Ulrich B. Hoff,

Johann Kollath-Bönig, Ulrik L. Andersen

QPIT, Dept. of Physics, Technical University of Denmark

Central European Workshop on Quantum Optics 2016

Kolymbari, Crete

Central European Workshop on Quantum Optics 2016

Kolymbari, Crete

live slides:  http://bit.ly/cewqo2016

"mechanical cats"

live slides:  http://bit.ly/cewqo2016

Goal (dream?):

to engineer macroscopic quantum states

of macroscopic objects

in particular:

a mechanical oscillator being in

a superposition of "here" and "there"

"mechanical cats"

live slides:  http://bit.ly/cewqo2016

UB Hoff et al., arXiv:1601.01663

Our proposal:

create and measure a mechanical superposition state using

  • cavity opto-mechanics
  • stroboscopic scheme for cooling, state prepration and read-out
  • initial non-Gaussian optical state

M. Aspelmeyer, T.J. Kippenberg, F. Marquardt,

Rev. Mod. Phys. 86, 1391 (2014)

live slides:  http://bit.ly/cewqo2016

\hat{H}_{\mathrm{int}} = -\hbar g_0 \hat{a}^\dagger\hat{a}(\hat{b}+\hat{b}^\dagger)
H^int=−ℏg0a^†a^(b^+b^†)\hat{H}_{\mathrm{int}} = -\hbar g_0 \hat{a}^\dagger\hat{a}(\hat{b}+\hat{b}^\dagger)
\hat{H}_{\mathrm{int}} \approx -\hbar 2 g_0 \sqrt{n_{\mathrm{cav}}} \frac{\delta\hat{a} + \delta\hat{a}^\dagger}{\sqrt{2}} \frac{\hat{b}+\hat{b}^\dagger}{\sqrt{2}} - \hbar \sqrt{2} g_0 n_{\mathrm{cav}} \frac{\hat{b}+\hat{b}^\dagger}{\sqrt{2}}
H^int≈−ℏ2g0ncavδa^+δa^†2b^+b^†2−ℏ2g0ncavb^+b^†2\hat{H}_{\mathrm{int}} \approx -\hbar 2 g_0 \sqrt{n_{\mathrm{cav}}} \frac{\delta\hat{a} + \delta\hat{a}^\dagger}{\sqrt{2}} \frac{\hat{b}+\hat{b}^\dagger}{\sqrt{2}} - \hbar \sqrt{2} g_0 n_{\mathrm{cav}} \frac{\hat{b}+\hat{b}^\dagger}{\sqrt{2}}
\hat{a} = \alpha + \delta \hat{a}, \quad \alpha = \sqrt{n_{\mathrm{cav}}}
a^=α+δa^,α=ncav\hat{a} = \alpha + \delta \hat{a}, \quad \alpha = \sqrt{n_{\mathrm{cav}}}
\hat{x}_L \hat{x}_M
x^Lx^M\hat{x}_L \hat{x}_M
\hat{x}_M
x^M\hat{x}_M

nonlinear, but weak interaction

use strong driving field and linearize

image/svg+xml

QND interaction

backaction (displacement)

optomechanical interaction

live slides:  http://bit.ly/cewqo2016

QND interaction

\omega_M \ll \tau^{-1} \ll \kappa
ωM≪τ−1≪κ\omega_M \ll \tau^{-1} \ll \kappa

evolution for                           

 + quadrature measurement

 + feedback

M. Vanner et al., PNAS 108, 16182 (2011)

pulsed optomechanics

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\hat{x}_L \rightarrow \hat{x}_L
x^L→x^L\hat{x}_L \rightarrow \hat{x}_L
\hat{p}_L \rightarrow \hat{p}_L + \chi \hat{x}_M
p^L→p^L+χx^M\hat{p}_L \rightarrow \hat{p}_L + \chi \hat{x}_M
\hat{x}_M \rightarrow \hat{x}_M
x^M→x^M\hat{x}_M \rightarrow \hat{x}_M
\hat{p}_M \rightarrow \hat{p}_M + \chi \hat{x}_L + \Omega
p^M→p^M+χx^L+Ω\hat{p}_M \rightarrow \hat{p}_M + \chi \hat{x}_L + \Omega
\tilde{p}_L
p~L\tilde{p}_L
\hat{x}_M -g\tilde{p}_L =-\hat{p}_L / \chi \quad \mathrm{for} \ g=\chi^{-1}
x^M−gp~L=−p^L/χfor g=χ−1\hat{x}_M -g\tilde{p}_L =-\hat{p}_L / \chi \quad \mathrm{for} \ g=\chi^{-1}
\rightarrow
→\rightarrow

brief pulse when oscillator in max. position - momentum kick 

(unresolved sideband regime)

pulsed scheme

live slides:  http://bit.ly/cewqo2016

1. Cooling of one quadrature

2. State transfer

3. Mechanical state tomography

M. Vanner et al., PNAS 108, 16182 (2011)

3 wishes at once!

pulsed scheme

live slides:  http://bit.ly/cewqo2016

V_{x_M}^{c} = \frac{V_{x_M}}{1+\chi_c^2 V_{x_M}/V_{p_L}}
VxMc=VxM1+χc2VxM/VpLV_{x_M}^{c} = \frac{V_{x_M}}{1+\chi_c^2 V_{x_M}/V_{p_L}}
\hat{x}_L \rightarrow \hat{x}_L
x^L→x^L\hat{x}_L \rightarrow \hat{x}_L
\hat{p}_L \rightarrow \hat{p}_L + \chi \hat{x}_M
p^L→p^L+χx^M\hat{p}_L \rightarrow \hat{p}_L + \chi \hat{x}_M
\hat{x}_M \rightarrow \hat{x}_M
x^M→x^M\hat{x}_M \rightarrow \hat{x}_M
\hat{p}_M \rightarrow \hat{p}_M + \chi \hat{x}_L + \Omega
p^M→p^M+χx^L+Ω\hat{p}_M \rightarrow \hat{p}_M + \chi \hat{x}_L + \Omega
\tilde{p}_L
p~L\tilde{p}_L
\hat{x}_M -g\tilde{p}_L =-\hat{p}_L / \chi
x^M−gp~L=−p^L/χ\hat{x}_M -g\tilde{p}_L =-\hat{p}_L / \chi
\rightarrow
→\rightarrow

1. Cooling of one quadrature

2. State transfer

3. Mechanical state tomography

pulsed scheme

\hat{x}_L \rightarrow \hat{x}_L
x^L→x^L\hat{x}_L \rightarrow \hat{x}_L
\hat{p}_L \rightarrow \hat{p}_L + \chi \hat{x}_M
p^L→p^L+χx^M\hat{p}_L \rightarrow \hat{p}_L + \chi \hat{x}_M
\hat{x}_M \rightarrow \hat{x}_M
x^M→x^M\hat{x}_M \rightarrow \hat{x}_M
\hat{p}_M \rightarrow \hat{p}_M + \chi \hat{x}_L + \Omega
p^M→p^M+χx^L+Ω\hat{p}_M \rightarrow \hat{p}_M + \chi \hat{x}_L + \Omega
\tilde{p}_L
p~L\tilde{p}_L
\rightarrow
→\rightarrow

1. Cooling of one quadrature

2. State transfer

3. Mechanical state tomography

\hat{x}_M -g\tilde{p}_L =-\hat{p}_L / \chi
x^M−gp~L=−p^L/χ\hat{x}_M -g\tilde{p}_L =-\hat{p}_L / \chi

live slides:  http://bit.ly/cewqo2016

\hat{x}_L \rightarrow \hat{x}_L
x^L→x^L\hat{x}_L \rightarrow \hat{x}_L
\hat{p}_L \rightarrow \hat{p}_L + \chi \hat{x}_M
p^L→p^L+χx^M\hat{p}_L \rightarrow \hat{p}_L + \chi \hat{x}_M
\hat{x}_M \rightarrow \hat{x}_M
x^M→x^M\hat{x}_M \rightarrow \hat{x}_M
\hat{p}_M \rightarrow \hat{p}_M + \chi \hat{x}_L + \Omega
p^M→p^M+χx^L+Ω\hat{p}_M \rightarrow \hat{p}_M + \chi \hat{x}_L + \Omega
\tilde{p}_L
p~L\tilde{p}_L
\rightarrow
→\rightarrow

pulsed scheme

1. Cooling of one quadrature

2. State transfer

3. Mechanical state tomography

\hat{x}_M -g\tilde{p}_L =-\hat{p}_L / \chi
x^M−gp~L=−p^L/χ\hat{x}_M -g\tilde{p}_L =-\hat{p}_L / \chi

live slides:  http://bit.ly/cewqo2016

input optical state

\hat{a}\hat{S}(r)|0\rangle
a^S^(r)∣0⟩\hat{a}\hat{S}(r)|0\rangle
|\alpha\rangle - |{-\alpha}\rangle\ ,\ \alpha=1
∣α⟩−∣−α⟩ , α=1|\alpha\rangle - |{-\alpha}\rangle\ ,\ \alpha=1

core tool for hybrid CV-DV QIP 

photon-subtracted squeezed vacuum state ("kitten")

UL Andersen, JSNN, Pv Loock, A Furusawa, Nat. Phys. 11, 713 (2015)

reviews:  JSNN et al., Prog. Inf. 8, 5 (2011)

live slides:  http://bit.ly/cewqo2016

model

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Mostly Gaussian:

  • initial light state (squeezed)
  • initial mechanics state (thermal)
  • photon subtraction beamsplitter
  • driving field displacement
  • optical losses
  • QND interaction
  • homodyne readout

Only photon detection is non-Gaussian

Model everything with covariance matrices and displacement vectors,

finalize with integration over detector's Wigner function 

Si3N4 tethered membrane

D. Kleckner et al. (2011)

R.A. Norte et al. (2015)

\omega_M/2\pi \approx 100\ \mathrm{kHz}
ωM/2π≈100 kHz\omega_M/2\pi \approx 100\ \mathrm{kHz}
Q_M \approx 10^8
QM≈108Q_M \approx 10^8
m_{\mathrm{eff}} \approx 1\ \mathrm{ng}
meff≈1 ngm_{\mathrm{eff}} \approx 1\ \mathrm{ng}
\kappa / 2\pi \approx 1\ \mathrm{GHz}
κ/2π≈1 GHz\kappa / 2\pi \approx 1\ \mathrm{GHz}
L \approx 4\ \mathrm{\mu m}
L≈4 μmL \approx 4\ \mathrm{\mu m}
g_0/2\pi \approx 400\ \mathrm{kHz}
g0/2π≈400 kHzg_0/2\pi \approx 400\ \mathrm{kHz}

micro-mirror on fibre tip

D. Hunger et al. (2010)

N.E. Flowers-Jacobs et al. (2012)

\chi \approx 1
χ≈1\chi \approx 1
\Omega \approx 400
Ω≈400\Omega \approx 400

optical pulse

\tau^{-1} \approx 10\ \mathrm{MHz}
τ−1≈10 MHz\tau^{-1} \approx 10\ \mathrm{MHz}
N_P \approx 3 \times 10^5
NP≈3×105N_P \approx 3 \times 10^5

simulated states

live slides:  http://bit.ly/cewqo2016

m = 1

m = 2

m = 3

with 1 thermal phonon

with 5% optical loss

10 dB squeezing

80% detector eff.

macroscopicity

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\mathcal{I} = -\frac{\pi}{2} \int \!\!\!\int dx dp \,W(x,p) \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial p^2} + 2\right) W(x,p)
I=−π2∫∫dxdpW(x,p)(∂2∂x2+∂2∂p2+2)W(x,p)\mathcal{I} = -\frac{\pi}{2} \int \!\!\!\int dx dp \,W(x,p) \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial p^2} + 2\right) W(x,p)

C-W. Lee, H. Jeong,

PRL 106, 220401 (2011)

quantify magnitude and frequency of interference fringes

macroscopicity

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effect of phase and amplitude noise in the displacement beam

what about dissipation?

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mechanical period:

rethermalization time scale:

decoherence time scale:

10 µs

~10 ms  (for 100 mK environment)

~100 µs  (for a cat state of size α = 4)

- so just about possible to prepare and read out state within one mechanical cycle

conclusion

live slides:  http://bit.ly/cewqo2016

Using

  • nonclassical light
  • pulsed, optomechanical coupling
  • homodyne detection and feedback
  • optimistic, but not unrealistic parameters

it should be possible to bring a mechanical oscillator into a cat state.

UB Hoff et al., arXiv:1601.01663

CEWQO 2017

at DTU, Lyngby

CEWQO 2017

at DTU, Lyngby

CEWQO 2017

at DTU, Lyngby

CEWQO 2017

at DTU, Lyngby

CEWQO 2017

at DTU, Lyngby

CEWQO 2017

at DTU, Lyngby

CEWQO 2017

at DTU, Lyngby

CEWQO 2017

at DTU, Lyngby

CEWQO 2017

at DTU, Lyngby

CEWQO 2017

at DTU, Lyngby

CEWQO 2017

at DTU, Lyngby

CEWQO 2017

at DTU, Lyngby

see you at

see you at

related work:

F. Khalili et al., PRL 105, 070403 (2010)

P. Sekatski et al., PRL 112, 080502 (2014)

R. Ghobadi et al., PRL 112, 080503 (2014)

J. Bennett et al., arXiv 1510.05368

A. Rakhubovsky et al., arXiv 1511.08611

...

quantum states of mechanics

live slides:  http://bit.ly/cewqo2016

manuscript:  arXiv:1601.01663

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