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Time
Coupling
Spatially adiabatic frequency conversion
in opto-electro-mechanical arrays
Ondřej Černotík,1,2
Sahand Mahmoodian,1
and Klemens Hammerer1
1
Institute for Theoretical Physics, Institute for Gravitational Physics (Albert Einstein Institute), Leibniz University Hannover, Germany
2
Max Planck Institute for the Science of Light, Erlangen, Germany
References Acknowledgements
This work was supported by the European Comission
(FP-7 Programme) through iQUOEMS (Grant Agreement
No. 323924). We gratefully acknowledge support by DFG
through QUEST and by the cluster system team at the
Leibniz University Hannover.
[1] OČ et al., arXiv:1707.03339.
[2] R. Andrews et al., Nature Physics 10, 321 (2014).
[3] L. Tian, Phys. Rev. Lett. 108, 153604 (2012).
[4] Y.-D. Wang and A. Clerk, Phys. Rev. Lett. 108, 153603 (2012).
Email: ondrej.cernotik@mpl.mpg.de
Summary
Optomechanical transduction
We show that an array of opto-electro-mechanical transducers can be used to enhance
the bandwidth of frequency conversion between microwaves and light [1]. In the array,
a mechanically dark mode of the propagating fields adiabatically changes from
microwave to optical or vice versa. Apart from the enhanced conversion bandwidth,
this strategy also leads to significantly reduced thermal noise. Remarkably, the
bandwidth enhancement per transducer element is largest for small arrays.
Conversion of itinerant fields Adiabatic passage
Transducer arrays
Losses and noise
Electromagnetic losses Thermal mechanical noise
0
1
2
3
Addednoise(unitsofshotnoise)
(a)
0.2 0.1 0.0 0.1 0.2
Frequency /
0.0
0.1
0.2
(b)
Bright mode
Dark mode
Cavity opto- and electromechanics
In a standard optomechanical system, the
motion of a mechanical oscillator modifies the
resonance of a cavity mode. The dynamics
can be described by the Hamiltonian
The weak dispersive interaction can be linearized around a strong
laser drive. When the drive frequency is detuned from the cavity
resonance by the mechanical frequency, , the cavity
field and the mechanical oscillator exchange excitations (provided
the system is in the resolved sideband regime ),
M. Aspelmeyer et al., RMP 86, 1391 (2014).
The same effects can also be observed in
electromechanical systems where the motion
of a capacitor plave couples to the field of an
LC circuit.
Conversion of propagating fields is possible in an
impedance-matched transducer, [2]. Such
a device can convert arbitrary input signals within the
bandwidth . Thermal mechanical noise is
suppressed for strong optomechanical cooperativity,
[3,4].
By coupling a mechanical oscillator to an optical cavity and
a microwave resonator, we can build a system for
converting signals between the two frequency domains [2].
The total interaction Hamiltonian is
Better bandwidth (equal to the cavity
linewidth) is possible with adiabatic
passage [3,4]. The signal is stored in one of
the cavities and the coupling rates are
varied so the signal remains in the
mechanically dark mode, .
Advantages of both approaches can be
combined in an array of optomechanical
transducers. The coupling rates are varied
adiabatically in space, enabling conversion of
arbitrary signals over a large bandwidth.
100
101
102
103
Array size N
0.0
0.2
0.4
0.6
0.8
1.0
Conversion|T21()|2
(a)
0.00 0.05 0.10 0.15 0.20
Frequency / R
(b)
Direct loss (on resonance) Backscattering
1.0 0.5 0.0 0.5 1.0
Frequency /
0.0
0.2
0.4
0.6
0.8
1.0
Conversion|T21()|2
(a)
100
101
102
103
Array size N
0.1
1
Bandwidth/
(b)
/ /
15
15
Phase
Far off resonance, the probability of converting a photon is
small, . In an array, it is enhanced -
fold, . We can therefore expect the bandwidth to
grow with the cubic root of the array size. A rigorous
derivation shows that for large arrays [1]
Conversion in small arrays (with ) can be
numerically optimized showing that bandwidth
enhancement is possible also in this regime.
Transfer matrix formalism
The dynamics of a single transducer is described
by the Heisenberg–Langevin equations which we
write in the matrix form
The equations can be solved in the frequency
domain. The effect of the transducer is described
by the scattering matrix
Signal propagation through the array can be
obtained by multiplying the scattering matrices of
individual transducers,
When the cavities decay through both ports, the
relationship between the scattering and transfer
matrices is more complicated. A transformation
between the two can, however, be found.
…
The transfer matrix formalism can also be used to
include free propagation between two transducers,
including propagation loss. Mechanical noise can be
added as a source term in the scattering
process, .
The electromagnetic fields can decay via two distinct
processes: direct loss (during propagation or in cavities)
and backscattering at mirrors. The latter has no
analogue in the usual adiabatic passage and leads to
interference between the incoming and reflected
signals.
Overall, the noise density is suppressed by the
collective cooperativity [1].
For small arrays, one can evaluate the added noise
numerically and show that it is also reduced
compared to using a single transducer.
To quantify the noise from the mechanical
reservoirs, we evaluate the associated spectral
density,
independent reservoirs
adiabaticitysingle transducer

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Spatially adiabatic frequency conversion in opto-electro-mechanical arrays

  • 1. Time Coupling Spatially adiabatic frequency conversion in opto-electro-mechanical arrays Ondřej Černotík,1,2 Sahand Mahmoodian,1 and Klemens Hammerer1 1 Institute for Theoretical Physics, Institute for Gravitational Physics (Albert Einstein Institute), Leibniz University Hannover, Germany 2 Max Planck Institute for the Science of Light, Erlangen, Germany References Acknowledgements This work was supported by the European Comission (FP-7 Programme) through iQUOEMS (Grant Agreement No. 323924). We gratefully acknowledge support by DFG through QUEST and by the cluster system team at the Leibniz University Hannover. [1] OČ et al., arXiv:1707.03339. [2] R. Andrews et al., Nature Physics 10, 321 (2014). [3] L. Tian, Phys. Rev. Lett. 108, 153604 (2012). [4] Y.-D. Wang and A. Clerk, Phys. Rev. Lett. 108, 153603 (2012). Email: ondrej.cernotik@mpl.mpg.de Summary Optomechanical transduction We show that an array of opto-electro-mechanical transducers can be used to enhance the bandwidth of frequency conversion between microwaves and light [1]. In the array, a mechanically dark mode of the propagating fields adiabatically changes from microwave to optical or vice versa. Apart from the enhanced conversion bandwidth, this strategy also leads to significantly reduced thermal noise. Remarkably, the bandwidth enhancement per transducer element is largest for small arrays. Conversion of itinerant fields Adiabatic passage Transducer arrays Losses and noise Electromagnetic losses Thermal mechanical noise 0 1 2 3 Addednoise(unitsofshotnoise) (a) 0.2 0.1 0.0 0.1 0.2 Frequency / 0.0 0.1 0.2 (b) Bright mode Dark mode Cavity opto- and electromechanics In a standard optomechanical system, the motion of a mechanical oscillator modifies the resonance of a cavity mode. The dynamics can be described by the Hamiltonian The weak dispersive interaction can be linearized around a strong laser drive. When the drive frequency is detuned from the cavity resonance by the mechanical frequency, , the cavity field and the mechanical oscillator exchange excitations (provided the system is in the resolved sideband regime ), M. Aspelmeyer et al., RMP 86, 1391 (2014). The same effects can also be observed in electromechanical systems where the motion of a capacitor plave couples to the field of an LC circuit. Conversion of propagating fields is possible in an impedance-matched transducer, [2]. Such a device can convert arbitrary input signals within the bandwidth . Thermal mechanical noise is suppressed for strong optomechanical cooperativity, [3,4]. By coupling a mechanical oscillator to an optical cavity and a microwave resonator, we can build a system for converting signals between the two frequency domains [2]. The total interaction Hamiltonian is Better bandwidth (equal to the cavity linewidth) is possible with adiabatic passage [3,4]. The signal is stored in one of the cavities and the coupling rates are varied so the signal remains in the mechanically dark mode, . Advantages of both approaches can be combined in an array of optomechanical transducers. The coupling rates are varied adiabatically in space, enabling conversion of arbitrary signals over a large bandwidth. 100 101 102 103 Array size N 0.0 0.2 0.4 0.6 0.8 1.0 Conversion|T21()|2 (a) 0.00 0.05 0.10 0.15 0.20 Frequency / R (b) Direct loss (on resonance) Backscattering 1.0 0.5 0.0 0.5 1.0 Frequency / 0.0 0.2 0.4 0.6 0.8 1.0 Conversion|T21()|2 (a) 100 101 102 103 Array size N 0.1 1 Bandwidth/ (b) / / 15 15 Phase Far off resonance, the probability of converting a photon is small, . In an array, it is enhanced - fold, . We can therefore expect the bandwidth to grow with the cubic root of the array size. A rigorous derivation shows that for large arrays [1] Conversion in small arrays (with ) can be numerically optimized showing that bandwidth enhancement is possible also in this regime. Transfer matrix formalism The dynamics of a single transducer is described by the Heisenberg–Langevin equations which we write in the matrix form The equations can be solved in the frequency domain. The effect of the transducer is described by the scattering matrix Signal propagation through the array can be obtained by multiplying the scattering matrices of individual transducers, When the cavities decay through both ports, the relationship between the scattering and transfer matrices is more complicated. A transformation between the two can, however, be found. … The transfer matrix formalism can also be used to include free propagation between two transducers, including propagation loss. Mechanical noise can be added as a source term in the scattering process, . The electromagnetic fields can decay via two distinct processes: direct loss (during propagation or in cavities) and backscattering at mirrors. The latter has no analogue in the usual adiabatic passage and leads to interference between the incoming and reflected signals. Overall, the noise density is suppressed by the collective cooperativity [1]. For small arrays, one can evaluate the added noise numerically and show that it is also reduced compared to using a single transducer. To quantify the noise from the mechanical reservoirs, we evaluate the associated spectral density, independent reservoirs adiabaticitysingle transducer