Jonas Neergaard-Nielsen PRO
Associate Professor @ DTU Physics, Denmark
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}} \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≈−ℏ2g0ncav2δa^+δa^†2b^+b^†−ℏ2g0ncav2b^+b^†
\hat{a} = \alpha + \delta \hat{a}, \quad \alpha = \sqrt{n_{\mathrm{cav}}}
a^=α+δa^,α=ncav
\hat{x}_L \hat{x}_M
x^Lx^M
\hat{x}_M
x^M
nonlinear, but weak interaction
use strong driving field and linearize
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
evolution for
+ quadrature measurement
+ feedback
M. Vanner et al., PNAS 108, 16182 (2011)
pulsed optomechanics
live slides: http://bit.ly/cewqo2016
\hat{x}_L \rightarrow \hat{x}_L
x^L→x^L
\hat{p}_L \rightarrow \hat{p}_L + \chi \hat{x}_M
p^L→p^L+χx^M
\hat{x}_M
\rightarrow \hat{x}_M
x^M→x^M
\hat{p}_M
\rightarrow \hat{p}_M + \chi \hat{x}_L + \Omega
p^M→p^M+χx^L+Ω
\tilde{p}_L
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
\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=1+χc2VxM/VpLVxM
\hat{x}_L \rightarrow \hat{x}_L
x^L→x^L
\hat{p}_L \rightarrow \hat{p}_L + \chi \hat{x}_M
p^L→p^L+χx^M
\hat{x}_M
\rightarrow \hat{x}_M
x^M→x^M
\hat{p}_M
\rightarrow \hat{p}_M + \chi \hat{x}_L + \Omega
p^M→p^M+χx^L+Ω
\tilde{p}_L
p~L
\hat{x}_M -g\tilde{p}_L =-\hat{p}_L / \chi
x^M−gp~L=−p^L/χ
\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{p}_L \rightarrow \hat{p}_L + \chi \hat{x}_M
p^L→p^L+χx^M
\hat{x}_M
\rightarrow \hat{x}_M
x^M→x^M
\hat{p}_M
\rightarrow \hat{p}_M + \chi \hat{x}_L + \Omega
p^M→p^M+χx^L+Ω
\tilde{p}_L
p~L
\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
live slides: http://bit.ly/cewqo2016
\hat{x}_L \rightarrow \hat{x}_L
x^L→x^L
\hat{p}_L \rightarrow \hat{p}_L + \chi \hat{x}_M
p^L→p^L+χx^M
\hat{x}_M
\rightarrow \hat{x}_M
x^M→x^M
\hat{p}_M
\rightarrow \hat{p}_M + \chi \hat{x}_L + \Omega
p^M→p^M+χx^L+Ω
\tilde{p}_L
p~L
\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
live slides: http://bit.ly/cewqo2016
input optical state
\hat{a}\hat{S}(r)|0\rangle
a^S^(r)∣0⟩
|\alpha\rangle - |{-\alpha}\rangle\ ,\ \alpha=1
∣α⟩−∣−α⟩ , α=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
live slides: http://bit.ly/cewqo2016
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
\omega_M/2\pi \approx 100\ \mathrm{kHz}
ωM/2π≈100 kHz
Q_M \approx 10^8
QM≈108
m_{\mathrm{eff}} \approx 1\ \mathrm{ng}
meff≈1 ng
\kappa / 2\pi \approx 1\ \mathrm{GHz}
κ/2π≈1 GHz
L \approx 4\ \mathrm{\mu m}
L≈4 μm
g_0/2\pi \approx 400\ \mathrm{kHz}
g0/2π≈400 kHz
\chi \approx 1
χ≈1
\Omega \approx 400
Ω≈400
optical pulse
\tau^{-1} \approx 10\ \mathrm{MHz}
τ−1≈10 MHz
N_P \approx 3 \times 10^5
NP≈3×105
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
live slides: http://bit.ly/cewqo2016
\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=−2π∫∫dxdpW(x,p)(∂x2∂2+∂p2∂2+2)W(x,p)
C-W. Lee, H. Jeong,
PRL 106, 220401 (2011)
quantify magnitude and frequency of interference fringes
macroscopicity
live slides: http://bit.ly/cewqo2016
effect of phase and amplitude noise in the displacement beam
what about dissipation?
live slides: http://bit.ly/cewqo2016
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
1+\frac{1}{1+x^2}
1+1+x21
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
Macroscopic mechanical superposition states
By Jonas Neergaard-Nielsen
Macroscopic mechanical superposition states
CEWQO 2016, Kolymbari
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