Microwave Photodetection with ElectroOptoMechanical Systems
Abstract
While detection of optical photons is today achieved with very high efficiencies, the detection of microwave fields at the photon level still poses nontrivial experimental challenges. In this Letter we propose a model of microwave photodetector which is based on the use of an electrooptomechanical system. Using state of the art technology, we show how single microwave photons can efficiently be converted into optical photons which are then measured by standard optics. The overall quantum efficiency of our microwave detector tends to one for achievable values of the optical and microwave cooperativity parameters. Our scheme could be used for sensing and scanning very faint microwave fields, as they may occur in radio astronomy, satellite and deep space communications.
pacs:
07.07.Df, 03.67.a, 03.65.w, 42.50.pIntroduction. Detection of photons is required for many fundamental and technological purposes. It is particularly relevant in the quantum regime, where the detection of few photons must be resolved, for both photocounting and access to the statistics of radiation field states (1); (2); (3). In the optical, near optical, and infrared domain, photodetection it is readily available through semiconductor technologies, where single photon resolution has been achieved. However microwave frequencies of the radiation are not directly reachable due to technological limitations. Several applications in cavity quantum electrodynamics (QED) have employed Rydberg atoms population detection as an indirect way of determining the coherence of FabryPerrot microwave cavity fields, but inevitably the atoms detection induce backaction over the field states (4).
Nowadays a large effort has been devoted to circuit QED, where transmission line resonators are coupled to superconducting devices, known as ‘artificial atoms’. Similarly to the FabryPerot cavities, indirect detection of charge or flux populations of the artificial atoms is employed as a way to infer physical properties of the radiation field (5). Unfortunately, the access to the statistical properties is only available when one can also access the number of microwave photons. Recent proposals for microwave photodetection using circuit QED have been demonstrated with limited quantum efficiency considering nowadays available superconducting technology (6); (7); (8); (9).
Moreover it would be interesting to have a photodetector able to work on demand in many different experimental situations and not restricted to a specific setup, similarly to the photodetectors used in the optical domain. One possible strategy to achieve this goal and circumvent both the limited detection efficiency and the problem of detection backaction is to perform a conversion of microwave photons into optical ones, where efficient photondetection is readily available. This is indeed the strategy suggested in our Letter.
We propose a microwave photodetector which is based on the efficient conversion of microwave photons into optical. This is realized by using an electrooptomechanical system where a mechanical oscillator induces strong coupling between a microwave and an optical resonator field. Under specific detuning of both the microwave and optical pumping fields, the average number of optical photons in the signal output is directly proportional to the mean number of microwave photons in the signal input. The proportionality parameter is an effective quantum efficiency which depends on optical and microwave cooperativity parameters associated with the system.
Adopting achievable experimental parameters for the electrooptomechanical transducer, we show that the efficiency of the microwaveoptical conversion can be extremely high. Combining this feature with the use of optical photodetection on the converted optical photons, we can achieve an overall quantum efficiency which is close to 100%. In our proposal there is no specific requirement for the input microwave signal, so that the setup can be used for microwave photon detection under several circumstances. In particular, it could be used for sensing faint microwave fields at the photon or subphoton level, with potential applications which include the environmental scanning of electrical circuits, radio astronomy, satellite and deep space communications.
System. As sketched in Fig. 1, the system consists of two microwave and optical cavity modes, coupled to a single mechanical resonator (11); (12); (13), with frequency and damping rate . The microwave (optical) cavity works at frequency with the total cavity decay rate being . We include the intrinsic losses with rates for both optical and microwave cavity modes, while denote the coupling rates of the respective input ports.
The optical and microwave cavity photons interact with the phononic modes of the mechanical via radiation pressure forces. This interaction is described by the following Hamiltonian (11); (10)
(1)  
where is the annihilation operator for mechanical resonator, is the annihilation operator for the cavity , and is the single photon electrooptomechanical coupling rate between cavity and mechanical resonator. Here we also assume that the microwave and optical cavities are driven at the frequencies , where is the detuning of the cavity and are the amplitudes of the driven pumps.
We can linearize the previous Hamiltonian by expanding the cavity modes around the steady state field amplitudes in each resonator. This is equivalent to set , where are the strength of the pumps, expressed in terms of mean number of cavity photons induced by the microwave and optical pumps (10); (14). The effective Hamiltonian of the system, in an interaction picture with respect to the free Hamiltonian, is therefore given by
(2) 
where are manyphoton optomechanical couplings.
In the Hamiltonian of Eq. (2) we have set the cavity detunings to be and assumed the regime of fast mechanical oscillations, so that we are in the resolved sideband regime for both cavities, with red sideband driving for both microwave and optical cavities. In this regime we have neglected the fast oscillating terms proportional to . The first term in Eq. (2) describes a beamsplitter like interaction between the mechanical resonator and propagating optical fields in the fibre at a rate . Similarly the second term describes the coherent exchange of the excitations between the mechanical resonator and the cavity microwave field at a rate . Note that the process of the exchange of excitations is coherent, as long as () (14); (15), where is Boltzmann constant, and is the temperature of the electrooptomechanical converter.
By using quantum Langevin equations (17) and standard inputoutput theory (1), the output variable of the optical cavity is given by
(3)  
where the coefficients , , , and satisfy and depend on the cooperativity terms (see (18) for details).
Pulse conversion. We assume the incoming microwave pulse signal is a coherent pulse with nonzero frequency spread . We choose such a coherent pulse to have its center frequency at the bare resonance frequency of the microwave cavity, i.e., . Therefore, the photon flux per unit angular frequency (power spectrum) is given by , where
with providing the total number of photons in the input pulse via . We see that is Gaussian (19) with standard deviation .
From Eq. (3) we can compute the power spectrum of the field at the output of the optical cavity, i.e., . This is given by
which depends on the microwave spectrum and
corresponding to Markovian noise added by the conversion process. Here and represent the thermal numbers of the mechanical and microwave baths, respectively, while is the thermal occupation number for the intracavity optical field. The mean photon number of the output optical field is equal to .
In order to see the effect of the converter on the shape of the incoming pulse, we plot in Fig. 2 both the incoming (microwave) pulse and converted (optical) pulse with respect to the normalized frequency . In this figure we have taken experimentalachievable parameters for the electrooptomechanical converter (14); (16) which is assumed to operate at cryogenic temperatures (K). We see how a faint microwave pulse ( photons) is successfully converted into an output optical pulse.
To analyze the general efficiency of the converter, we consider the ratio between the output optical photons and the input microwave photons, i.e., . This ratio depends on the values of the two cooperativity parameters and . As we can see from Fig. 3, the larger these parameters are, the better is the microwave to optical conversion, which rapidly approaches the ideal conversion rate of 100.
Quantum efficiency of the microwave detector. At the output of the converter, the optical photons are measured by a photodetector with high efficiency and large bandwidth (up to 10GHz, i.e., much larger than ). Assuming an incoming microwave pulse with small bandwidth (so that we can approximate the Gaussian pulse with a delta function), the mean number photons which are detected at the optical output is equal to
(4) 
where is the quantum efficiency of the microwave detector, and .
From Eq. (4) we can see that the mean number of detected optical photons has a term which is proportional to the mean number of photons which were present in the input microwave pulse. This is equivalent to having a beamsplitter with transmissivity mixing the incoming microwave field with vacuum fluctuations. Then there is an additional term which accounts for the Markovian thermalization occurring in the conversion process, and depending on the thermal baths of the mechanical oscillator and intracavity microwave field.
Working at cryogenic temperatures (e.g., K), the thermalization term can be neglected and the effective quantum efficiency of the microwave detector is indeed , where accounts for the strength of the coupling between the optical and microwave fields with the mechanical resonator, which is monotonically increasing in the two cooperativity terms. Assuming a highefficient optical detector and other achievable experimental conditions, the quantum efficiency can approach 100 for sufficiently high values of the cooperativity parameters, as shown in Fig. 4.
Conclusions. We have designed a model of microwave photodetector whose working mechanism is based on the use of an electrooptomechanical converter. In our approach, microwave fields with a small number of photons can efficiently be converted into optical fields, which are then subject to standard optical measurements. We have shown that overall quantum efficiency of our detector can be very high assuming cryogenic temperatures and achievable experimental parameters. Our receiver can potentially be used in all those scenarios connected with the detection of faint microwave signals, including deep space communications and radio astronomy, for instance, in the mapping of the cosmic background radiation.
Acknowledgments. S.B. is grateful for support from the Alexander von Humboldt foundation. M.C.O. acknowledges support from CNPq/FAPESP through the Instituto Nacional de Ciência e Tecnologia em Informação Quântica (INCTIQ) and FAPESP through the Research Center in Optics and Photonics (CePOF). S.P. has been supported by a Leverhulme Trust research fellowship and EPSRC (via ‘qDATA’, grant no. EP/L011298/1).
Note added. Upon completion of our manuscript, we noted that a similar work by Zhang et al. has been recently submitted to the arxiv (arXiv:1410.0070).
References
 D.F. Walls and Gerard J. Milburn, Quantum Optics, (2ï¿½ed., Ed. Springer, 2007).
 Marlan O. Scully and M. Suhail Zubairy, Quantum optics, (Cambridge University Press, 1997).
 Y. Yamamoto and A. Imamoglu. Mesoscopic Quantum Optics, (Wiley, 1999).
 S. Haroche and J.M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons, (Oxford Univ. Press, New York, 2006).
 M. H. Devoret and R. J. Schoelkopf, Science 339, 1169 (2013).
 G. Romero, J. J. GarciaRipoll, and E. Solano, Phys. Rev. Lett. 102, 173602 (2009).
 B. Peropadre, G. Romero, G. Johansson, C. M. Wilson, E. Solano, and J. J. GarciaRipoll, Phys. Rev. A 84, 063834 (2011).
 A. Poudel, R. McDermott, and M. G. Vavilov, Phys. Rev. B 86, 174506 (2012).
 B. Fan, G. Johansson, J. Combes, G. J. Milburn, and T. M. Stace, Phys. Rev. B 90, 035132 (2014).
 Sh. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, Phy. Rev. A 84, 042342 (2011).
 Sh. Barzanjeh et al., Phys. Rev. Lett, 109, 130503 (2012).
 R. W. Andrews et al., Nature Phy, 10, 321 (2014).
 J. Bochmann et al., Nature Phy. 9, 712 (2013).
 T. A. Palomaki et al., Science 342, 710 (2013).
 S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, Phys. Rev. A 84, 052327 (2011).
 T. A. Palomaki, J. W. Harlow, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, Nature 495, 210 (2013).

The dynamics of the system can be described by following quantum Langevin equations

The explicit expressions of the coefficients are
 C. Weedbrook, S. Pirandola, R. GarcíaPatrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys. 84, 621 (2012).