Optomechanical cooling of levitated spheres with doubly-resonant fields
Optomechanical cooling of levitated dielectric particles represents a promising new approach in the quest to cool small mechanical resonators towards their quantum ground state. We investigate two-mode cooling of levitated nanospheres in a self-trapping regime. We identify a rich structure of split sidebands (by a mechanism unrelated to usual strong-coupling effects) and strong cooling even when one mode is blue detuned. We show the best regimes occur when both optical fields cooperatively cool and trap the nanosphere, where cooling rates are over an order of magnitude faster compared to corresponding single-sideband cooling rates.
Extraordinary progress has been made in the last half-dozen years Review (); Kipp () towards the final goal of cooling a small mechanical resonator down to its quantum ground state and hence to realise quantum behavior in a macroscopic system. Implementations include cavity cooling of micromirrors on cantilevers Metzger (); Arcizet (); Gigan (); Regal (), dielectric membranes in Fabry Perot cavities membrane (); radial and whispering gallery modes of optical microcavities Schliess () and nano-electromechanical systems NEMS (). Indeed the realizations span 12 orders of magnitude Kipp (), up to and including the LIGO gravity wave experiments. Corresponding advances in the theory of optomechanical cooling have also been made Brag (); Paternostro (); Marquardt (); Wilson ().
Over the last year or so, a promising new paradigm has been attracting much interest: several groups Isart (); Zoller (); Barker (); Ritsch () have now proposed schemes for optomechanical cooling of levitated dielectric particles, including nanospheres and even viruses Isart (). The important advantage is the elimination of the mechanical support, a dominant source of heating noise. In general, these proposals involve two fields, one for trapping and one for cooling. This may involve an optical cavity mode plus a separate trap; or two optical cavity modes, the so-called “self-trapping” scenario.
Mechanical oscillators in the self-trapping regime differ from other optomechanically-cooled devices in a second fundamental respect (in addition to the absence of mechanical support): the mechanical frequency, , associated with centre of mass oscillations is not an intrinsic feature of the resonator but is determined by the optical field. In particular, it is a function of one or both of the detuning frequencies, and , of the optical modes. Cooling, in general, occurs when is resonantly red detuned with either of the detuning frequencies (i.e. negative is associated with cooling). For self-trapping systems, this means so the relevant frequencies are not independent.
The full implications of this nonlinear interdependence of the resonant frequencies have not yet been fully elucidated. We show here for the first time that it leads to a rich landscape of split sidebands. The mechanism here is unrelated to splittings seen in experiments in the strong-coupling regime strong (). However, it results in extremely favourable cooling regimes, where two (or more) cooling sidebands approach each other. We term this the “double-resonance” regime. We find it can produce cooling rates nearly two orders of magnitude stronger than the corresponding “single-resonance” case.
Two optical field modes are coupled to a nanosphere with centre of mass position . is given in the rotating frame of the laser which drives the modes with amplitudes and respectively. In Zoller (), the phase between the optical potentials was chosen to be ; the study focussed primarily on the regime, where the mode is responsible exclusively for trapping while the mode alone provides cooling. Previous studies Isart (); Zoller (); Ritsch () all analysed mechanical oscillations about an equilibrium position , corresponding to the antinode of the trapping mode (field 1). Below, this scenario is referred to as the “single-resonance” regime.
Here we investigate the effects of relaxing all these restrictions and find interesting and unexpected implications. We take ; the cooling field is driven more weakly than the trapping field, but with a ratio . Below, our analytical expressions cover arbitrary , but we compare with an illustrative set of experimentally plausible parameters: we take a cavity damping Hz. We considered driving powers in the range mW, where . For a laser of wavelength nm and a cavity of length cm, waist we consider a silica nanosphere of nm radius and hence a coupling strength Hz. To obtain a dephasing of between the two modes near the centre of the cavity, the frequency difference between the modes is GHz. This far exceeds the detunings MHz and also the mechanical frequencies . Thus the photons are completely distinguishable and can be read out and driven separately. Nevertheless, since Hz, we approximate . However, this situation is distinct from the ring-cavity proposal of Ritsch () where and mode 2 is undriven but is populated exclusively by scattering from mode 1; there, the photons are of precisely the same frequency and thus there is a single detuning parameter involved.
Fig.2 illustrates the behavior for . It shows that allowing both fields to cooperatively trap and cool yields more than an additive improvement. We denote by and the set of detunings corresponding to cooling resonances of modes 1 and 2 respectively: Fig.2 shows that these resonances unexpectedly split and separate into new cooling resonances and . These can overlap, to give very strong cooling associated with multiple resonances. Below, we also show that the usually studied single-field resonance regime can attain only a maximal cooling rate ; and that for strong driving, : thus, the cooling falls with increasing driving and it is hard to achieve optimal cooling regimes where . For the double-field resonances, in contrast, , the cooling increases with and can more easily reach optimal cooling. Very strong cooling is apparent even for regimes where one mode is blue detuned. In addition, although there is no direct coupling between optical modes, double-resonances offer the prospect of strong (albeit second-order) coupling and entangling of the two modes via the nanosphere, within a single cavity. This includes simultaneous resonant/antiresonant regimes (indicated by the crossing of and in Fig.2) where one mode resonantly heats, while the other resonantly cools the mechanical mode.
The dynamics depends on . However transforming to scaled variables reduces this complexity. We rescale position, time and field variables as follows: , , then Note that below we drop all the tildes but it is implicit that all variables are scaled in the resulting Heisenberg equations:
The dynamics for a given depends only on the the scaled driving where , two scaled detunings and a scaled damping ; all scaled frequencies (including cooling rates) are given below as a fraction of .
The experimentally adjustable parameters are , both the detunings and . We assume , though the analytical expressions are for arbitrary . Varying driving power mW , but leaving the cavity/nanosphere properties unchanged means remains constant, but varies from .
Following the usual procedure, we replace operators by their expectation values and linearise about equilibrium fields by performing the shifts , and . Hence we find equilibrium photon fields, and as well as position .
Here, and . The dimensionless mechanical frequency is:
Closely related forms of this “self-trapping” frequency expression have been noted previously Isart (); Zoller (); Barker (); Ritsch () but the implications, other than for , have not been investigated.
To first order, the linearised equations of motion are:
where . From the above, we can obtain the contribution from the two photon fields to the optomechanical cooling:
(net cooling occurs for ). We also calculate a numerical by evolving the equations of motion in time and looking at the decay in (its variance in particular). The analytical cooling rates give excellent agreement with numerics in all but the strongest cooling regions.
The single-field cooling resonance occurs for , and , thus here for (i.e. ). Conversely, there is also a single-field cooling resonance for (note that since we consider only the case , i.e. field 2 is always driven more weakly than field 1, the latter situation does not correspond simply to an interchange in the role of the fields). Away from these extreme cases, both cooling resonances are split by the effect of the other field, wherever .
From Fig.2 we see occur for the same , thus the same equilibrium photon field ; however they correspond to photon fields and respectively and thus to different , where is the splitting (about ) between and seen in Fig.2.
The transformation leaves both the mechanical frequency and unchanged. Hence the cooling rates are similar for both .
We can estimate the splitting by requiring
Close to , we can simplify . Similarly, , the splitting between is .
We now analyse the relative merits of single-resonance versus double-resonance cooling. Single-field cooling corresponds to in Fig.2. Cooling rates are obtained from Eq.5 by taking , and . This regime was investigated in Zoller () and Eq.5 reduces to expressions therein (in unscaled units). However, we can give good approximations to cooling rates purely in terms of experimental parameters (driving power, and ). Assuming and that the field 1 contribution to cooling is negligible, near , the mechanical frequency . Hence, as shown in the appendix, the Single Resonance (SR) cooling rate becomes:
(recall this is a scaled cooling rate thus given in units of ).
Single-field cooling is a maximum if where (in unscaled units) and is thus at the edge of the resolved sideband regime. Here, ; this gives optimal cooling only if . This cooling maximum is independent of : it depends only on . As the driving is increased, if ,
Thus the single resonance cooling rate falls off quite rapidly as the driving amplitude is increased: the cooling cannot be improved by increasing the driving amplitude.
To obtain the corresponding double-resonant rate (DR) one must first identify the -dependent pair of detunings for which . Even if do not cross in Fig.2, the sidebands approach within their width and overlap significantly. One can still obtain a good approximation to the cooling rate in terms of driving parameters (see Appendix). In this case, there are contributions to cooling from both field 1 and field 2. Adding them both,
where the frequency is given by the expression, . The contribution from mode 2 while that of mode 1 ; both contribute significantly for . Assuming , this reduces to , hence .
Fig.4 shows that Eqs.9 and Eqs.11 both give excellent agreement with
exact numerics. In the double-resonant case, the cooling is stronger and
increases with increasing . In contrast, the single-resonant cooling
and this cannot be improved by increasing . Self-trapping cooling
cannot be considered simply in terms of an additive contribution from two intracavity
intensities in Eq.6; the response of to the driving is also
important. In the double-resonant case,
for strong driving. In contrast, for
the single-resonant case and strong driving pushes the
resonance into the far-detuned regime.
A study of the quantum cooling shows that the minimum phonon numbers attainable
for single and double resonant cooling for strong driving
. For weak driving, strong cooling can be obtained with
single field cooling, but at the edge of the resolved sideband regime ,
less favourable for ground state cooling.
Conclusion Our study shows that the two-mode self-trapping regime has a raft of of unexpected features, including the split side-bands, strong cooling at blue-detuning and simultaneous heating and cooling resonances. Although other proposals also permit strong cooling rates, the multiple sidebands provide an exceptionally broad region of strong cooling, offering considerable robustness to experimental errors in the driving power, detunings and even the phase (a variation of of order 30% will not appreciably perturb the strong cooling).
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Appendix A Appendix
a.1 Minimum phonon number
From quantum perturbation theory we can show that , the rate of transition from state n to n+1 is:
For , this gives the cooling rate of (Eq.5). However, with the exact expressions we can also show the equilibrium mean phonon number to be:
The parameters for strongest cooling do not necessarily yield the minimum phonon numbers, because the rate depends upon the difference between the optical heating and cooling, while the phonon number depends upon the ratio between the two. In Figs.5 and 6 we present colour maps comparing the cooling and minimum phonon numbers for both and respectively. Fig.5 corresponds to the same parameters as Fig.2. It shows the broad strong cooling region corresponding to the three distinct cooling resonances which are only partially resolved. The result is a strong cooling region of about 1MHz width, providing the advantage of a strong cooling regime insensitive to experimental detunings. For Fig.6 in contrast, the map has a high degree of symmetry, since the role of the modes is interchangeable; the larger means the splitting is larger, so are fully resolved at the double resonance: thus only contribute to the maximal cooling region. Nevertheless, this is the point where the strongest cooling () is obtained.
The figures show that the largest cooling rates are found in the symmetric double resonance regime (the double resonance for ), the phonon occupancy is slightly lower in the usually analysed Zoller () single-field regime where one strong field, at near zero detuning, traps while the other field cools. Nevertheless, the mean equilibrium occupancy is very small in both cases (less than a tenth of a phonon). The above analysis of minimum phonon occupancy has not considered other sources of heating (photon scattering, background gas collisions). The effect of these other heating rates has been analysed in Zoller () where ground state cooling is shown to be achievable for optimal cooling rates (). Thus the 1-2 order of magnitude increase in cooling in the double-resonance region means that this is the most favourable regime.
a.2 r2 and single field cooling
The single-field cooling rates are obtained from Eq.5 by taking , and .
The above, in principle, requires a full numerical solution of the equations:
to find equilibrium positions and fields. However we can obtain a good approximation in closed form using experimental parameters (driving power, detunings). Hence,
For resonant cooling and cooling/heating by field 1 is negligible. So,
and , hence the Single Resonance (SR) cooling rate becomes:
NB: this is a scaled cooling rate thus given in units of . This gives a maximum cooling rate if where
It is worth noting that this maximum is independent of : it depends only on . (Within the underlying assumption that ) Even for , the maximum cooling is far from optimal .
As the driving is increased, if ,
Thus the single resonance cooling rate falls off quite rapidly as the driving amplitude is increased: the optimal cooling cannot be attained by increasing the driving amplitude.
a.3 and overlap: double-resonance cooling
As illustrated in Fig.2, the resonances and never actually coincide: they undergo something reminiscent of an avoided crossing (recall that Fig.2 is a map of the classical cooling, not of underlying quantum eigenvalues. Nonetheless, the similarity between the classical linearisation and the effective Hamiltonian for the quantum fluctuations has a very similar structure.
The double-field cooling rates for the region of closest approach are estimated from Eq.5 by assuming that the two resonances actually cross, in other words both fields are, to a good approximation, simultaneously resonant.
The first task is to identify the -dependent pair of detunings for which:
One can search numerically for detunings which give near-simultaneous resonances. However, for moderate , a closed form can be obtained. Provided is small the double resonance still falls in the small angle regime (including our Fig.2). In this case, From Eq.16:
so if we obtain .
The mechanical frequency:
In this case, there are contributions to cooling from both field 1 and field 2. Adding them both,
where we will substitute Eq.25 to evaluate the frequency . As ever, :
Eqs.26 and 27 should be contrasted with the behaviour of the singly resonant cooling Eq.10. In the double-resonant case, the cooling increases, without limit, as a function of . For and (ie 8mW power), . In Fig.4 the approximate expressions Eqs.10,26,27 are shown to give quite good agreement with cooling rates obtained from numerical solution of the equations of motion, without linearisation.
The very large damping rates provided by the double-resonant regime have the added advantage of relative insensitivity to the initial preparation. Although the analysis assumes small oscillations about equilibrium , a resonator prepared at the antinode of the trapping field () is very rapidly pulled towards equilibrium and oscillates about . Consequently there is no need to provide any initial displacement. In addition, the multiple resonance region shown in Fig.2 is surprisingly robust to errors in the relative phase between the two modes. In Fig.5 we see that a phase error of order 20% makes relatively little difference (and may even enhance the cooling).
a.4 Symmetric double-resonance cooling
The best regime for strong cooling is one in which the fields are equally strong and where both are resonantly, red detuned. In this region:
In the limit :