Laser cooling and optical detection of excitations in a LC electrical circuit
We explore a method for laser cooling and optical detection of excitations in a LC electrical circuit. Our approach uses a nanomechanical oscillator as a transducer between optical and electronic excitations. An experimentally feasible system with the oscillator capacitively coupled to the LC and at the same time interacting with light via an optomechanical force is shown to provide strong electro-mechanical coupling. Conditions for improved sensitivity and quantum limited readout of electrical signals with such an “optical loud speaker” are outlined.
pacs:42.50.Wk, 78.20.Jq, 85.60.Bt
Cooling plays an essential roles in most areas of physics, in part because it reduces detrimental thermal fluctuations. For sensing application, where thermal fluctuations may hide the small signals one is trying to measure, strong coupling of mechanical and electrical oscillators to systems in a pure quantum state, such as light or polarized atomic ensembles, opens up new possibilities for quantum sensing of fields and forces braginsky92 (). This in principle allows for enhanced sensitivity of the oscillators, where readout of their state is limited only by quantum fluctuations. In recent years, dramatic advances in optomechanical coupling and cooling of high quality-factor (Q) mechanical systems have been made Kippenberg (); Aspelmeyer (); Tombesi (); Vahala (); Karrai (); Girvin ().
In this Letter, we propose to extend laser cooling of mechanical objects to electrical circuits. By coupling a high-Q inductor-capacitor resonator (LC) to a near-resonant nanomechanical membrane harrisXX (); Kimble () in the radio frequency (rf) domain, the electrical circuit can be effectively cooled by the cold mechanical system. Since such electrical circuits are used in a wide variety of settings, the reduction of thermal fluctuations in these system will likely find numerous applications. We also show that the cooling techniques explored here allow for optical readout of electrical signals in the circuit. Since light fields are routinely measured with quantum limited precision this allows for high sensitivity broadband detection of weak electric signals. Moreover, light and atomic ensembles can behave as oscillators with a negative mass and thus provide the means to measure fields and forces beyond the standard quantum limit using the power of entanglement Wasilewski (); Hammerer (); Caves (). Finally, our coupling occurs at the level of individual radio frequency photons; thus, for systems with sufficiently low thermal load, our approach provides a versatile interface for quantum information, allowing for the reliable transfer of quantum states from radio frequency to optical domains and back.
The key idea in this work is to achieve sufficient coupling between a nanomechanical membrane and a high-Q LC circuit to induce normal mode splitting simmonds10 (), and then observe the electrical excitations via opto-mechanical coupling between the membrane and a high-Q optical cavity. Our suggested method is insertion of the membrane into the fringing field of a capacitor silvan (), as shown in Fig. 1, such that the capacitance depends upon the displacement of the membrane. We demonstrate that for reasonable component parameters, when combined with a voltage bias and an inductive component to make a resonant electrical circuit near the frequency of a mechanical resonance , the coupling between radio-frequency (rf) photons and the membrane phonons can become sufficiently large to induce normal mode splitting, where the resonant response of the system comprises combined electro-mechancial excitations.
A model Hamiltonian describing the coupled electromechanical system (Fig. 1) in the limit of well separated, high-Q resonant electrical and mechanical modes is
Here , , and . The flux and the membrane momentum are the canonical momenta conjugate to the charge and position . When the two modes are brought into resonance , the natural canonical variables become normal mode solutions with ; ; ; , with frequencies for as considered below.
This coupled-mode system could be inserted into an optical cavity (Fig. 1), whose mode couples to the membrane position via radiation pressure harrisXX (). Both normal modes interact with the cavity mode as ; thus, radiation pressure-based detection can be applied independently to each normal mode when the cavity linewidth is narrow enough to resolve the normal mode splitting . Alternatively, both normal modes can be observed simultaneously when . In both cases the optical system can also cool the combined electro-mechanical system. In essence this cooling is achieved because the membrane-cavity system acts as a transducer up-converting excitations in the LC circuits to optical frequencies. This means that the LC circuit will equilibrate with the optical modes which are in the quantum mechanical ground state even at room temperature. At the same time the interface between rf excitations in the LC circuits and optical photons also allows for detection of rf-electric signals by optical measurement. Since optical measurement can by quantum noise limited this opens up new possibilities for detection weak electrical signals, as we outline below.
We provide a specific design which admits a strong coupling of a nanomechanical resonator to an LC circuit and simultaneously admits optical coupling (Fig. 1). Specifically we envision a parallel plate capacitor where we replace one of the plates by a set of wires of thickness , width , and separated by a distance , see Fig. 2(a). In order to have an interaction with light we introduce a hole in the capacitor to allow for laser beams to get through (not shown). This hole has a minimal effect on the capacitance when the membrane area is much larger than the mode cross section of the cavity, consistent with a (1 mm) membrane and a 50 m cavity waist. A dielectric membrane of thickness is inserted into the capacitor a distance from the top of the wires. The replacement of one of the capacitor plates by the set of wires creates a spatial inhomogeniety of the electric field, which attracts the membrane towards the wires when the capacitor is charged. At the same time this inhomogeniety also means that the capacitance will depend on the position of the membrane . Expanding the capacitance around the equilibrium position gives rise to an LC-membrane coupling . This is analogous to the radiation pressure coupling that occurs in the optical domain, i.e., a type nonlinearity. As such, we can enhance the coupling strength by providing a classical displacement of the LC circuit’s charge, with either a dc or ac voltage bias providing an offset charge, . For simplicity we restrict ourselves to the case of a dc voltage, though generalization to the ac case is a simple extension of these ideas and allows to frequency match the LC and mechanical systems. The coupling between the membrane position and the charge fluctuations around the equilibrium is then and is enhanced by the large charge induced on the capacitor, in direct analogy to the similar effect for cavity optomechanics. The full Hamiltonian including electrical and mechanical contributions is:
Here is the equilibrium membrane position at . The fixed point of the classical charge at a given bias voltage and the equilibrium displacement are then found from which yields
Here we have introduced a characteristic length scale defined by , which describes the relative change in the capacitance at the new equilibrium position .
Around these classical values, we consider the remaining quantum fluctuations . We change to annihilation and creation operators for the membrane (LC), and find a Hamiltonian
with . This corresponds to the model Hamiltonian examined in the beginning. Assuming a constant value of from to , the coupling constant can be expressed in a more intuitive form , i.e., through the capacitance change caused by the displacement of the membrane due to the applied voltage . The solution is a stable point under the condition . We have neglected a small nonlinear correction, due to the femtometer length scale of the zero-point membrane fluctuations. Devices which enhance this nonlinear coupling are of interest, but beyond the scope of the present work.
To obtain quantitative estimates of the feasible coupling constant, we assume the plate electrode to be much larger than the separation of the plate and transverse dimensions of the wires . We can then find the capacitance for a given position of the membrane by solving for the potential using the finite element method. We express the capacitance as , where is a dimensionless number of order unity, which describe the deviation from a standard parallel plate capacitor.
As an example, we take a SiN membrane of dielectric constant and thickness nm inserted into a capacitor with a separation m and dimensions , , and . A simulation with these values is shown in Fig. 2 (b). From this simulation we extract the values of m for a distance of m. Hence if the applied voltage shifts the equilibrium position by nm around m the coupling constant is and the system is in the strong coupling regime if the Q values of the LC circuit and membrane exceed 100. Assuming an operating frequency of MHz and an oscillator length fm this displacement only requires an applied voltage on the order of a few volts. If a larger initial separation is desirable, a similar coupling () could be achieved if the equilibrium distance is shifted from m to m ( m for m) with an applied voltage of several hundred volts.
Ii Cooling the LC circuit
The membrane may be efficiently cooled via optomechanical coupling between the radiation pressure force of a cavity field and the position of the central area of the membrane. The details of this process have been analyzed by a wide variety of groups Girvin (). In essence the effect on the membrane degree of freedom is to induce a damping , which is much greater than the intrinsic damping rate of the membrane , but is limited by the cavity decay rate . This additional damping only produces moderate additional quantum fluctuations associated with the vacuum noise of the light field (which can be accounted for by adjusting the temperature of defined below). Working with the LC circuit with damping , resonant with the membrane (), we may expect an efficient coupling between the optomechanical system and the LC circuit, provided that such that we can get excitations out of the system faster than they leak in.
We use the input-output formalism in the rotating wave approximation to find a full description of this combined mode cooling. The Heisenberg-Langevin equations describing this situation are
In a strong damping limit (), we can treat the coupled LC resonator as a perturbation and arrive at
This equation describes the cooling of through the membrane-light system with a rate . In the continuous cooling limit, we expect to achieve a thermal population in given by
where are the original thermal occupation of modes and . Typically this population will be dominated by the heating of the LC circuit (the first term) since the membrane can have a very large Kimble ().
In the mode-resolved, strong-coupling limit, with and , each of the two normal modes and have frequencies and a damping rate given given by the average of the two damping rates . The optomechanical coupling then works independently on each of the two modes. Assuming again the of the membrane to be much higher than the of the LC, a standard argument for optomechanical cooling marquardt08 () leads to a thermal occupation number of .
Comparing the two limits derived above we see that the minimal thermal occupation is achieved at a cooling laser power and detuning such that , where we obtain a population . The cooling of the membrane is, however, limited by the cavity decay rate . Hence the cooling limit is the larger of and . We have neglected optical heating effects, consistent with our assumption of good sideband-resolution ().
Iii Sensitivity of optical readout of LC circuit
The cooling identified above realizes an interface between optical fields and rf excitations of the LC circuits at the single photon level, and we now turn to a possible application of this interface. Often LC circuits are used in sensitive detectors to pick up very small signals NMRloop (). As we will now show, the sensitivity in such experiments can be improved by detecting the cooling light leaving the cavity. This takes advantage of the fact that the homodyne detection of laser light can be quantum noise limited with near-unit quantum efficiency, thus avoiding many of the noise sources present for low frequency signals. To show this we will work in the strong damping limit identified above with the LC circuit tuned into resonance with the membrane (). Again we also assume the damping of the mechanical motion of the membrane to be negligible . In this limit the membrane and the cavity mediate an effective interaction between the LC mode and the optical cavity input/output modes with the effective cooling rate introduced above. In the rotating wave approximation this situation is described by the generic equations
Here, we have introduced an incoming signal to be measured, which is described by . If the signal is from a voltage applied to the system, .
Suppose that we want to measure the Fourier components of the incoming signals detuned by a frequency with respect to the resonance frequency of the LC circuit within a certain bandwidth . This can be done by splitting the outgoing signal on a beamsplitter (, where is the annihilation operator for the other mode incident on the beamsplitter) and inferring the two quadratures and from a homodyne detection of the () quadrature of the () mode. The signal-to-noise ratio for, e.g., a measurement of the amplitude can be defined by . Here describes the noise in the absence of any signal. From the equations of motion we find
Here describes the number of thermal excitations in the field used to probe the circuit and we assume that the fields incident on the beamsplitter are of the same type such that .
Let us compare our approach to the case where the LC circuit is read out by homodyne detection with an rf amplifier assumed to have a similar number of thermal excitations as the system being measured . Disregarding any additional noise added during the amplification, is optimized for and is limited to with a detection bandwidth . In contrast, with the optical readout, the incoming laser fields can be quantum noise limited with if light is in a coherent state. In this case we obtain twice the signal-to-noise ratio for . The optimal sensitivity is thus better with optical detection, even if we assume ideal detection of the fields in both cases. Such an ideal detection is routinely achieved by homodyne detection of optical fields with near unity quantum efficiency, whereas it is hard to achieve for rf fields. For realistic limited detect or efficiencies of rf fields, the sensitivity may thus be significantly improved using optical readout. Furthermore the high sensitivity with laser cooling is obtained over a much larger bandwidth which is determined by . In other words, if, prior to laser cooling, the LC circuit had a high-Q and a narrow bandwidth that is less than the bandwidth required for a particular application, laser cooling allows an increase of the bandwidth with a limited decrease in the sensitivity ( dB) if . Using regular rf techniques, an alternative approach would be to increase the bandwidth by increasing the damping of the circuit, but this would result in a decrease of the sensitivity by a factor of . Crucially, since optical fields are shot-noise limited even at room temperature, this measurement setup does not involve cryogenics.
The potential benefits of this approach – high quantum efficiency conversion from radio frequency to optical photons, and the corresponding potential for low temperature detection of radio frequency signals – are limited by the finite Q values for room temperature inductors. Appropriate replacements may be considered, such as crystal resonators or cryogenic superconducting resonators. An additional benefit of a cryogenic setup is the possibility to enter the quantum strong coupling limit, , at which point the conversion from radio frequency to optical domain can be used as a quantum interface. However, understanding of these features and improvements requires further investigation.
We gratefully acknowledge useful discussions with Koji Usami, Ole Hansen, Silvan Schmidt, Anja Boisen, and John Lawall. JMT thanks the NBI for hospitality during his stay. This research was funded by ARO MURI award W911NF0910406, DARPA and by the EU project Q-ESSENCE.
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