Trajectories Without Quantum Uncertainties in Composite Systems with Disparate Energy Spectra

It is well established that measurement-induced quantum back action (QBA) can be eliminated in composite systems by engineering so-called quantum-mechanics-free subspaces (QMFSs) of commuting variables, leading to a trajectory of a quantum system without quantum uncertainties. This situation can be realized in a composite system that includes a negative-mass subsystem, which can be implemented by, e.g., a polarized spin ensemble or a two-tone-driven optomechanical system. The realization of a trajectory without quantum uncertainties implies entanglement between the subsystems, and allows for measurements of motion, fields, and forces with, in principle, unlimited precision. To date, these principles have been developed theoretically and demonstrated experimentally for a number of composite systems. However, the utility of the concept has been limited by the dominating requirement of close proximity of the resonance frequencies of the system of interest and the negative-mass reference system, and by the need to embed the subsystems in a narrowband cavity, which could be problematic while at the same time achieving good overcoupling. Here we propose a general approach that overcomes these limitations by employing periodic modulation of the driving fields (e.g., two-tone driving) in combination with coherent or measurement-based antinoise paths. This approach makes it possible to engineer a QMFS of two systems with vastly different spectra and with arbitrary signs of their masses, while dispensing with the need to embed the subsystems in a sideband-resolving cavity. We discuss the advantages of this novel approach for applications such as QBA evasion in gravitational wave detection, force sensing, and entanglement generation between disparate systems.

Popular Summary

In the current push to exploit the advances of quantum science for technological applications, there is a need to combine highly different quantum systems, such as cold or hot atoms, mechanical resonators, solid-state impurities, or superconducting circuits, in order to build hybrid devices with complex functionalities. This poses the challenge of efficiently linking systems that evolve with different characteristic frequencies and that are probed by electromagnetic radiation fields of different frequencies (e.g., light or microwaves). The present work proposes a general technique for overcoming those challenges by tailored drive fields and entangled beams of light.

The concept of using modulated drive fields to alter the response of a physical system has a long history in quantum metrology. The flip side of this approach is that it also alters the character of the measurement quantum back action (QBA) on the system, i.e., the stochastic disturbance imposed on the observed system by the measurement. Suppression of the extraneous QBA components allows for a joint measurement on a composite (hybrid) system in a so-called quantum-mechanics-free subspace (QMFS). Here we offer feasible methods for suppressing the modulation-induced extraneous QBA by either coherent cancelation or direct measurement and subtraction in postprocessing. With this in place, modulated driving can be used to establish a QMFS that effectively links the disparate constituents of distributed hybrid systems.

The ability to establish QMFSs between a variety of disparate physical platforms opens up a trove of new possibilities for quantum networking and sensing applications using hybrid quantum systems.

Figure 1

Basic single-pass topologies for quantum-noise-evading joint measurements on two localized oscillators (pendula; see Sec. 1 for candidate systems) by means of traveling meter fields (wiggly arrows). In the serial topology, the (stochastic) QBA forces $\propto {\hat{a}}_j^c$ (green dashed arrows) on the two localized oscillators are perfectly correlated ${\hat{a}}_2^c={\hat{a}}_1^c$ (in the absence of optical losses) simply due to the fact that the same itinerant field is probing the two subsystems. The QBA responses (thick arrows with green filling) of both oscillators are mapped into the phase quadrature ${\hat{b}}_2^s$ of the output light. These contributions will interfere destructively, assuming the counter-rotating frequency configuration ${\mathrm{\Omega }}_1=-{\mathrm{\Omega }}_2$ with matched light-oscillator coupling rates ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2$. In the parallel topology, a source of entangled light correlates the QBA forces $\mathrm{Cov}[{\hat{a}}_1^c,{\hat{a}}_2^c]>0$ of the two meter fields and anticorrelates the measurement imprecision noises $\mathrm{Cov}[{\hat{a}}_1^s,{\hat{a}}_2^s]<0$ so that both interfere destructively when the two output photocurrents are combined in postprocessing, again provided that ${\mathrm{\Omega }}_1=-{\mathrm{\Omega }}_2$ and ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2$.

Figure 2

Basic principle behind the universal frequency conversion technique as exemplified by two-tone probing (lower part) and juxtaposed with the case of conventional single-tone driving (upper part). These schemes for coupling an oscillator to a traveling light field are illustrated by their individual output spectra; the two-photon (homodyne) phase quadrature ${\hat{b}}_j^s(\mathrm{\Omega })$ results from folding the spectrum around the local oscillator frequency ${\omega }_{\mathrm{LO},j}={\omega }_{o,j}$. (top) Single-tone coupling of an oscillator with eigenfrequency ${\mathrm{\Omega }}_1$ using a tone at ${\omega }_{o,1}$ (solid line); the sidebands peak at frequencies of approximately ${\omega }_{o,1}\pm {\mathrm{\Omega }}_1$ (true in the high-$Q$ limit, $|{\mathrm{\Omega }}_j|/(2{\gamma }_j)\gg 1$; sideband widths are exaggerated in the figure for illustrative purposes). (bottom) Two-tone coupling scheme using coherent tones at ${\omega }_{o,2}\pm \tilde{\mathrm{\Omega }}$ (solid lines) with mean frequency ${\omega }_{o,2}$ (dashed line), resulting in the effective oscillator evolution frequency ${\mathrm{\Omega }}_{2,\mathrm{eff}}\equiv \mathrm{\Lambda }\mathrm{sign}{\mathrm{\Omega }}_2=(|{\mathrm{\Omega }}_2|-\tilde{\mathrm{\Omega }})\mathrm{sign}{\mathrm{\Omega }}_2$. The curved arrows indicate the sidebands generated by the respective drive tones in the particularly interesting case $\tilde{\mathrm{\Omega }}>|{\mathrm{\Omega }}_2|\iff \mathrm{\Lambda }<0$, where the sign of the evolution frequency is flipped in the down-conversion process, $\mathrm{sign}{\mathrm{\Omega }}_{2,\mathrm{eff}}=-\mathrm{sign}{\mathrm{\Omega }}_2$. This reflects the circumstance that, for $\tilde{\mathrm{\Omega }}>|{\mathrm{\Omega }}_2|$, the upper (lower) central sideband arises as the lower (upper) sideband of the originating drive tone. If a joint measurement is performed on oscillators 1 and 2, broadband destructive interference between the QBA responses contained in the green, central sidebands is possible if ${\mathrm{\Omega }}_1=-{\mathrm{\Omega }}_{2,\mathrm{eff}}$.

Figure 3

Spectral mappings for a two-tone-driven oscillator showing the emergence of the extraneous QBA sidebands. Both paths linking the nominal QBA (green dashed arrows) to the nominal output sidebands ${\hat{a}}^c(\mathrm{\Omega })\to {\hat{b}}^s(\mathrm{\Omega })$ (blue arrows with green filling) via $\hat{X}$ have net zero contribution from the phase $\mathrm{\Phi }$, in contrast to the extraneous QBA components (yellow dashed arrows) that are mixed into the nominal output ${\hat{a}}^c(\mathrm{\Omega }\pm 2\tilde{\mathrm{\Omega }})\to {\hat{b}}^s(\mathrm{\Omega })$ (blue arrows with yellow filling) with a net phase contribution of $e^{\mp 2i\mathrm{\Phi }}$. The extraneous output components ${\hat{b}}^s(\mathrm{\Omega }\pm 2\tilde{\mathrm{\Omega }})$ (black arrows with filling) can be filtered out in postprocessing (indicated by black shutters).

Figure 4

Scheme for removing extraneous QBA (27) by invoking an auxiliary measurement channel and postprocessing. Spectral separation of the output light field $\hat{b}$ from the oscillator is performed using an external narrowband cavity ($|\mathrm{\Omega }|\ll {\kappa }_{\mathrm{filter}}\ll \tilde{\mathrm{\Omega }}$) resonant at the optical carrier frequency ${\omega }_{\mathrm{filter}}={\omega }_o$. The resulting reflection of the high-frequency sideband components $|\mathrm{\Omega }|\gg {\kappa }_{\mathrm{filter}}$ of $\hat{b}(\mathrm{\Omega })$ (absolute frequencies ${\omega }_o\pm \mathrm{\Omega }$) permits a direct measurement of the stochastic, extraneous QBA force, arising from components $|\mathrm{\Omega }|\gtrsim \tilde{\mathrm{\Omega }}\gg {\kappa }_{\mathrm{filter}}$ (yellow dashed arrow) of the amplitude quadrature ${\tilde{b}}^c(\mathrm{\Omega })={\hat{a}}^c(\mathrm{\Omega })$ [in the particular case of two-tone driving, the relevant components are $\mathrm{\Omega }\sim \pm 2\tilde{\mathrm{\Omega }}$; see Eq. (36)]. The transmitted, low-frequency sideband components $|\mathrm{\Omega }|\ll {\kappa }_{\mathrm{filter}}\ll \tilde{\mathrm{\Omega }}$ of ${\hat{b}}^s(\mathrm{\Omega })$ (blue thick arrows), Eq. (25), are subjected to a standard phase quadrature measurement ${\tilde{b}}^s(\mathrm{\Omega })={\hat{b}}^s(\mathrm{\Omega })$. By postprocessing the auxiliary measurement current ${\tilde{b}}^c$ according to the oscillator response function in Eq. (27) and subtraction from that of ${\tilde{b}}^s$, the extraneous QBA contribution (blue arrow with yellow filling) is removed from the latter, resulting in an effective oscillator readout of the desired form, Eq. (24), containing only the nominal QBA contribution (blue arrow with green filling) induced by ${\hat{a}}^c(\mathrm{\Omega })$ (green dashed arrow).

Figure 5

Scheme for removing the extraneous QBA contribution (36) (thick arrows with yellow filling) induced by the spectral sideband components $\sim \pm 2\tilde{\mathrm{\Omega }}$ (yellow dashed arrow) of ${\hat{a}}_1^c$ using coherent cancelation. A single, effective oscillator with the desired behavior is formed from cascading two identical oscillators that are two-tone driven out of phase: the extraneous QBA responses (thick arrows with yellow filling) cancel, while the nominal QBA responses (thick arrows with green filling) interfere constructively.

Figure 6

Estimates of GWD position sensitivity $\sqrt{S^x}$ (71) improved by quantum noise evasion using an effective auxiliary oscillator with negative mass; see Eqs. (68) and (69). Left: absolute position sensitivity. Right: sensitivity gain $\sqrt{S_{\mathrm{std}}^x/S^x}$ relative to a GWD without a quantum-noise-reducing auxiliary system and without squeezing. The values of the parameters are those listed in Table 1 unless otherwise noted.