Measurement-induced macroscopic superposition
states in cavity optomechanics

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

Johann Kollath-Bönig, Amine Laghaout, Ulrik L. Andersen

QPIT, Dept. of Physics, Technical University of Denmark

Measurement-induced macroscopic superposition
states in cavity optomechanics

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

Johann Kollath-Bönig, Amine Laghaout, Ulrik L. Andersen

QPIT, Dept. of Physics, Technical University of Denmark

  • a few thoughts on macroscopicity
  • photon subtraction
  • micro-macro entanglement
  • macroscopic states of mechanical oscillators

outline

Schrödinger cat state

|awake\rangle+|asleep\rangle
∣awake⟩+∣asleep⟩|awake\rangle+|asleep\rangle
|\uparrow\rangle|awake\rangle+|\downarrow\rangle|asleep\rangle
∣↑⟩∣awake⟩+∣↓⟩∣asleep⟩|\uparrow\rangle|awake\rangle+|\downarrow\rangle|asleep\rangle
|alive\rangle+|dead\rangle
∣alive⟩+∣dead⟩|alive\rangle+|dead\rangle
|awake\rangle+|asleep\rangle
∣awake⟩+∣asleep⟩|awake\rangle+|asleep\rangle

(as told for children)

what is macroscopic?

  • many particles?
  • many modes?
  • large excitation?
  • distinguishable / detectable by macroscopic means?
  • ...

phase space macroscopicity

\mathcal{N} = \langle n_{\mathrm{fluctuations}} \rangle = \langle n \rangle - \langle n_{\mathrm{displacement}} \rangle
N=⟨nfluctuations⟩=⟨n⟩−⟨ndisplacement⟩\mathcal{N} = \langle n_{\mathrm{fluctuations}} \rangle = \langle n \rangle - \langle n_{\mathrm{displacement}} \rangle
= \frac{1}{2} (\mathrm{Var}(x)+\mathrm{Var}(p)-1)
=12(Var(x)+Var(p)−1)= \frac{1}{2} (\mathrm{Var}(x)+\mathrm{Var}(p)-1)

define "size" of pure state as

ideally, "macroscopicity" should also tell something about the macroscopic distinguishability or "distance" between component states,

\mathcal{M} = \mathcal{N} \times \mathcal{D} \quad?
M=N×D?\mathcal{M} = \mathcal{N} \times \mathcal{D} \quad?

A. Laghaout, JSNN, U.L. Andersen, Opt. Comm. 337, 96 (2015)

\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)

generalisation to mixed states:

quantify magnitude and frequency of interference fringes

photon subtraction

\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

Powerful tool for optical state manipulation

photon-subtracted squeezed vacuum state ("kitten")

kittens in the wild (2012)

many other uses, e.g. entanglement distillation

H. Takahashi et al., Nat. Phot. 4, 178 (2010)

JSNN et al., Progress in Informatics 8, 5 (2011)

reviews

U.L. Andersen, JSNN, P. van Loock, A. Furusawa, Nat. Phys. 11, 713 (2015)

\sqrt{T}\hat{a}_A+\sqrt{1-T}\hat{a}_B
Ta^A+1−Ta^B\sqrt{T}\hat{a}_A+\sqrt{1-T}\hat{a}_B

U.L. Andersen, JSNN, PRA 88, 022337 (2013)

micro-macro entanglement

DV/CV or micro-macro entanglement by ambiguous photon subtraction

Morin et al., Nat. Phot. 8, 570 (2014)

related experiments

Jeong et al., Nat. Phot. 8, 564 (2014)

\sqrt{T}\hat{a}_A+\sqrt{1-T}\hat{a}_B
Ta^A+1−Ta^B\sqrt{T}\hat{a}_A+\sqrt{1-T}\hat{a}_B
|1\rangle_A \ \hat{S}_B(r)|0\rangle_B \rightarrow |0\rangle_A \ \sqrt{T} \hat{a}^m \hat{S}_B(r)|0\rangle_B + |1\rangle_A \ \sqrt{1-T} \hat{a}^{m+1} \hat{S}_B(r)|0\rangle_B
∣1⟩A S^B(r)∣0⟩B→∣0⟩A Ta^mS^B(r)∣0⟩B+∣1⟩A 1−Ta^m+1S^B(r)∣0⟩B|1\rangle_A \ \hat{S}_B(r)|0\rangle_B \rightarrow |0\rangle_A \ \sqrt{T} \hat{a}^m \hat{S}_B(r)|0\rangle_B + |1\rangle_A \ \sqrt{1-T} \hat{a}^{m+1} \hat{S}_B(r)|0\rangle_B

U.L. Andersen, JSNN, PRA 88, 022337 (2013)

|1\rangle_A \ \hat{S}_B(r)|0\rangle_B \rightarrow |0\rangle_A \ \sqrt{T} \hat{a}^m \hat{S}_B(r)|0\rangle_B + |1\rangle_A \ \sqrt{1-T} \hat{a}^{m+1} \hat{S}_B(r)|0\rangle_B
∣1⟩A S^B(r)∣0⟩B→∣0⟩A Ta^mS^B(r)∣0⟩B+∣1⟩A 1−Ta^m+1S^B(r)∣0⟩B|1\rangle_A \ \hat{S}_B(r)|0\rangle_B \rightarrow |0\rangle_A \ \sqrt{T} \hat{a}^m \hat{S}_B(r)|0\rangle_B + |1\rangle_A \ \sqrt{1-T} \hat{a}^{m+1} \hat{S}_B(r)|0\rangle_B
\frac{1}{2}\big(|0\rangle_A+|1\rangle_A\big)\Big| \quad\quad\quad\Big\rangle_B + \frac{1}{2}\big(|0\rangle_A-|1\rangle_A\big)\Big| \quad\quad\quad\quad\ \ \Big\rangle_B
12(∣0⟩A+∣1⟩A)∣⟩B+12(∣0⟩A−∣1⟩A)∣  ⟩B\frac{1}{2}\big(|0\rangle_A+|1\rangle_A\big)\Big| \quad\quad\quad\Big\rangle_B + \frac{1}{2}\big(|0\rangle_A-|1\rangle_A\big)\Big| \quad\quad\quad\quad\ \ \Big\rangle_B

U.L. Andersen, JSNN, PRA 88, 022337 (2013)

use rotated basis:

x-quadrature distributions

\Big| \quad\quad\quad \big(x\big) \Big|^2
∣(x)∣2\Big| \quad\quad\quad \big(x\big) \Big|^2
\Big| \quad\quad\quad\quad\ \ \big(x\big) \Big|^2
∣  (x)∣2\Big| \quad\quad\quad\quad\ \ \big(x\big) \Big|^2

U.L. Andersen, JSNN, PRA 88, 022337 (2013)

size

distinguishability

(success of homodyne discrimination)

size & distinguishability - comparison

homodyne detector

photon number detector

A. Laghaout, JSNN, U.L. Andersen, Opt. Comm. 337, 96 (2015)

\hat{a}\hat{S}(r)|0\rangle \pm \hat{a}^2 \hat{S}(r)|0\rangle
a^S^(r)∣0⟩±a^2S^(r)∣0⟩\hat{a}\hat{S}(r)|0\rangle \pm \hat{a}^2 \hat{S}(r)|0\rangle
|\alpha\rangle \pm |{-\alpha}\rangle
∣α⟩±∣−α⟩|\alpha\rangle \pm |{-\alpha}\rangle
\hat{D}(\alpha)\frac{|0\rangle \pm |1\rangle}{\sqrt{2}}
D^(α)∣0⟩±∣1⟩2\hat{D}(\alpha)\frac{|0\rangle \pm |1\rangle}{\sqrt{2}}

cat

"kitten"

displaced Fock states

mechanical kittens

Can we prepare a mechanical oscillator in a macroscopic superposition state?

M. Aspelmeyer, T.J. Kippenberg, F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014)

\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 cavity field and linearize

pulsed interaction: couple only to single quadrature

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

B. Julsgaard et al., Nature 432, 482 (2004)

QND interaction

state transfer

- as in e.g. atomic quantum memory

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

eqs. of motion for                                  

(bad cavity regime)

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

 + quadrature measurement

 + feedback

pulsed interaction

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

protocol

prepare optical state

displace

pulsed QND interaction

measure & feed-forward

U.B. Hoff, J. Kollath-Bönig, JSNN, U.L. Andersen, arXiv:1601.01663

U.B. Hoff, J. Kollath-Bönig, JSNN, U.L. Andersen, arXiv:1601.01663

U.B. Hoff, J. Kollath-Bönig, JSNN, U.L. Andersen, arXiv:1601.01663

Si3N4 tethered membrane

D. Kleckner et al. (2011)

R.A. Norte et al. (2015)

\omega_M/2\pi = 100\ \mathrm{kHz}
ωM/2π=100 kHz\omega_M/2\pi = 100\ \mathrm{kHz}
Q_M = 10^8
QM=108Q_M = 10^8
m_{\mathrm{eff}} = 1\ \mathrm{ng}
meff=1 ngm_{\mathrm{eff}} = 1\ \mathrm{ng}
\kappa / 2\pi = 1\ \mathrm{GHz}
κ/2π=1 GHz\kappa / 2\pi = 1\ \mathrm{GHz}
L = 4\ \mathrm{\mu m}
L=4 μmL = 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 = 1
χ=1\chi = 1
\Omega = 400
Ω=400\Omega = 400

optical pulse

\tau^{-1} \approx 10\ \mathrm{MHz}
τ−1≈10 MHz\tau^{-1} \approx 10\ \mathrm{MHz}
N_P \approx 3.2 \times 10^5
NP≈3.2×105N_P \approx 3.2 \times 10^5
r=1.15 \ (10\ \mathrm{dB}) \quad T=0.98 \quad\bar{n}_{\mathrm{th}}=1
r=1.15 (10 dB)T=0.98n¯th=1r=1.15 \ (10\ \mathrm{dB}) \quad T=0.98 \quad\bar{n}_{\mathrm{th}}=1
\eta=0.98
η=0.98\eta=0.98
\eta=0.50
η=0.50\eta=0.50

on/off

detector

m=0
m=0m=0
m=1
m=1m=1
m=2
m=2m=2
m=3
m=3m=3

total negativity

\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} dx_M dp_M |W_M^{out}(x_M,p_M)|-1
∫−∞∞∫−∞∞dxMdpM∣WMout(xM,pM)∣−1\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} dx_M dp_M |W_M^{out}(x_M,p_M)|-1

PNR

effect of splitting ratio and detector efficiency

on/off

macroscopicity vs. initial thermal occupation

pre-cooling & tomography

3 pulses:

  • use an initial QND measurement to cool one mechanical quadrature
  • use a final QND measurement to read out a quadrature

M. Vanner et al., Nat. Comm. 4, 2295 (2013)

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

final state sensitive to displacement amplitude (e.g. laser intensity fluctuations)

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

...

summary

  • squeezing and photon subtracted squeezing: 

          simple optical kitten state generation

          micro/macro entanglement

  • feasible transfer to mechanical kittens by QND interaction

slides at http://slides.com/jneer/badhonnef-2016

Ulrik Lund Andersen

Amine Laghaout

Ulrich Busk Hoff

slides at http://slides.com/jneer/badhonnef-2016

Johann Kollath-Bönig