An efficient cooling of the quantized vibration by a four-level configuration
Cooling vibrational degrees of freedom down to ground states is essential to observation of quantum properties of systems with mechanical vibration. We propose two cooling schemes employing four internal levels of the systems, which achieve the ground-state cooling in an efficient fashion by completely deleting the carrier and first-order blue-sideband transitions. The schemes, based on the quantum interference and Stark-shift gates, are robust to fluctuation of laser intensity and frequency. The feasibility of the schemes is justified using current laboratory technology. In practice, our proposal readily applies to an nano-diamond nitrogen-vacancy center levitated in an optic trap or attached to a cantilever.
pacs:42.50.Wk, 07.10.Cm, 37.10.-x, 03.65.Yz
Introduction.- To observe quantum characteristics of some systems with spin-vibration coupling, e.g., trapped ions or nanomechanical cantilevers, we have to eliminate the thermal phonons intrinsically owned by the systems. As such, cooling a system down to ground states of the vibrational degrees of freedom is usually a prerequisite of executing quantum tasks, such as quantum computing rmp-75-281 () and ultra-precision measurements nature-430-329 (); pnas-106-1313 (); prl-108-120801 (). Despite great success so far in cooling of trapped ions or atoms, cooling solid-state systems down to vibrational ground states is still tough experimentally.
Heating during the cooling process originates from two aspects. One is from the environment. This heating in solid-state systems can be suppressed by reducing the surface area of the system (or say, enhancing the quality factor Q), decreasing the work temperature of the surrounding environment nature-459-960 (); nature-464-697 () and/or dynamically controlling the dissipation prl-110-153606 (). The other is from some unexpected processes in the cooling, such as the carrier and the blue-sideband transitions. Suppressing those undesired transitions is an effective way to the cooling acceleration prl-85-4458 () . As such, modified cooling schemes using quantum interference, such as the electromagnetic induced transparency (EIT), can largely suppress the blue-sideband transitions nano-cooling-exp (); nature-443-193 (); prl-85-4458 () and work more efficiently than the original idea of sideband cooling prl-99-093901 (); prl-99-093902 (); nano-cooling-theory (); rmp-86-1391 (). In addition, a proposal involving Stark-shift gate njp-9-279 () can suppress both carrier and blue-sideband transitions by steering the system to red-sideband transitions.
In the present letter, we propose two efficient cooling schemes by suppressing the unexpected processes as mentioned above, through employing four internal levels of the systems. As clarified below, the schemes can readily apply to vibrational systems involving nano-diamond nitrogen-vacancy (nNV) centers. Due to large mass and special characteristic, the nNV center system cannot be cooled down by simply merging previous cooling ideas, such as the scheme with EIT plus Stark-shift-gate for cooling trapped ions prl-104-043003 (). Magnetic field gradient, in addition to laser irradiation, is required to couple the internal to the vibrational degrees of freedom of the solid-state system. Besides, due to existence of an additional decay to a metastable level in the nNV center nature-466-730 (), a pumping process in addition to usual cooling operations is demanded oe-21-029695 (). Even with all these considerations, however, a three-level structure employed in a nNV center is proven to be not qualified for a perfect cooling since the first-order blue sideband transition cannot be fully eliminated oe-21-029695 (); SR-5-14977 (); prb-79-041302 ().
In contrast, the two schemes proposed in the present letter employ four internal levels of the nNV center, which can well accomplish the cooling by completely eliminating the carrier and the first-order blue sideband transitions. One of our schemes with a modified -type configuration is based on a dynamical EIT prl-85-4458 (); oe-21-029695 () plus a Stark-shift gate njp-9-279 (); SR-5-14977 (), which is called briefly as the asymmetric cooling method. The other, called shortly as the symmetric cooling method, combines the -type with -type configurations, which yield double Stark-shift gates. As we know, most of the cooling schemes proposed so far are based on -type three-level systems prl-85-4458 (); prl-103-227203 (); oe-21-029695 (); rmp-75-281 (), rather than the -type three-level configuration prb-79-041302 () , due to the fact that the latter with two upper levels is more susceptible to dissipation. However, as shown below, a better cooling could be achieved if we elaborately combine the -type and -type configurations and have them interfered with each other. The interference enhanced by the Stark-shift gates yields a dark state and a Stark-shift-gate point, which help for an efficient cooling.
The systems. - We exemplify two systems to clarify our schemes. One is an nNV center levitated in an optic trap nphtonics-9-653 () (see Fig. 1(a)), which is promising for detecting quantized gravity pra-90-033834 (), preparing large distance superpositions prl-107-020405 (); pra-88-033614 () and building matter-wave interferometers prl-111-180403 (). The other is the nNV center attached to a cantilever nphy-7-879 () (see Fig. 1(b)), which has potential applications in observation of phononic Mollow triplet natcomm-6-8603 (), ultra-sensitive measurements prl-108-120801 (), quantum information processing prl-105-220501 (); natphys-6-602 () and biological sensing nnano-3-501 (). To achieve the objectives, the vibrational degrees of freedom in those systems are required to be cooled down to ground states which are very challenging with current laboratory technologies. We show below that our schemes can accomplish the cooling of the above systems in an efficient and robust way.
The nNV center owns multiple levels and a small surface area, which ensure it to be operated more flexibly and with lower heating from the environment than other solid-state candidates. Besides, the internal electronic states of the nNV center and the corresponding vibrational states can be coupled strongly under a modest magnetic field gradient ( T/m) science-335-1603 (). Provided that the trapping frequency of the nNV center (or the vibrational frequency of the cantilever) is and a magnetic field gradient along the nNV center axis couples the nNV electron spin () to its vibration, such a system can be described in units of as , where () is the annihilation (creation) operator of the vibration, is the energy difference between induced by the magnetic field and the coupling strength is prb-79-041302 () with the zero-point fluctuation amplitude , the mass of the NV center , the Lande factor , the Bohr magneton , and the magnetic field gradient along the nNV center axis. The ground state with represents the Zeeman sublevels of the spin . Due to intrinsic spin-spin coupling properties, there exists a zero-field splitting of GHz between and science-314-281 (); prl-112-047601 (); prb-85-205203 (); njp-13-025025 ().
The asymmetric cooling method. - The nNV center is driven by external light fields as in Fig. 1(c). To carry out the cooling scheme, we first employ a dark state, which, under the condition of , is . Correspondingly, we define the states and . The steady state, obtained from the first-order expansion of the parameter prl-104-043003 () can be written as
where () denotes the phonon state. Normalization factor is omitted in Eq. (1) for simplicity, but considered in calculations below. The effective Hamiltonian contributing to the cooling is with
where with . denotes the free energy term, and and are terms employed in the EIT cooling and the Stark-shift-gate cooling, respectively. So it is clear that the asymmetric cooling method works with a combination of the EIT cooling and the Stark-shift-gate cooling. For the steady state , we have , which remains invariant under the condition
This condition, also called Stark-shift-gate point, implies that all the terms in act on the steady state by a destructive interference and thus the steady state does not suffer from any loss due to spontaneous emission.
The symmetric cooling method. - With respect to the asymmetric cooling method, we apply an additional microwave to couple the ground state to the ground state , as shown in Fig. 1(d). The level configuration in this case is actually a combination of the -type and -type structures, which looks graphically symmetric. In this case, it is easily proven that is the dark state under the double resonant conditions. The Hamiltonian here is written as , with
where the state . Eq. (4) indicates that the symmetric cooling method works with two Stark-shift-gate cooling processes collaboratively. The first-order expansion of for the steady state (omitting the normalization factor) prl-104-043003 () is . Similar to the asymmetric cooling method, we can have a simplified expression from applied on , which is invariant once the detuning meets the condition of Stark-shift-gate point,
The cooling effects. - We first solve the cooling coefficients by the fluctuation spectrum, which is given by spra-46-2668 ()
where implies the expectation value over the steady state. In the Schrödinger representation, the operator with and . For the steady state in the asymmetric cooling scheme, we find two forces, and , contributing to the cooling. These two forces split the fluctuation spectrum into three parts, that is, , and their interaction term supplementary information (). The heating coefficient can be obtained by the fluctuation spectrum as with supplementary information (). Under the condition in Eq. (3), we have and zero values in fluctuation spectra at the points (see Fig. 2(a)), which means that quantum interference induced by the EIT and the Stark-shift gate fully eliminates heating from the first-order blue sideband and carrier transitions. At the detuning , the cooling coefficient reaches the peak value
which is similar to the square-law style of the cooling laser strength in the EIT cooling scheme oe-21-029695 ().
Similar to the asymmetric cooling scheme, the fluctuation spectra in symmetric counterpart are obtained and plotted in Fig. 2(b). The heating coefficient in this scheme reaches zeros if Eq. (5) is satisfied. When we choose , a maximal cooling coefficient is obtained as
To demonstrate the cooling effects more specifically, we simulate the dynamics of the system under a zero-temperature environment by the Lindblad master equation where
with . We find working regions of different cooling schemes, as sketched in Fig. 3a. Although the asymmetric cooling scheme favors a weaker coupling (i.e., smaller values of ), stronger lasers (i.e., larger values of ) are necessary in that case. As plotted in Fig. 3b, the asymmetric scheme can achieve the lowest final phonon number among the four typical schemes when 0.05. In contrast, if we have a stronger coupling (e.g., with bigger magnetic field gradient), we may achieve the cooling by the symmetric scheme using weaker lasers (see Fig. 3c). An evident reason for this difference between the two schemes is due to different forms of as in Eqs. (7) and (8) as well as the relation of the cooling coefficient with the final phonon number, i.e., in the case of . Deeper physics for this difference is reflected in different properties between the EIT cooling and the Stark-shift-gate cooling. In the asymmetric scheme, the -type structure plus the Stark-shift gate constitutes an enhanced EIT cooling that is steered to a better cooling rate under the condition of . In contrast, the symmetric cooling is essentially an enhanced Stark-shift cooling, in which the better cooling occurs if . Since is always smaller than 0.5, stronger lasers are definitely required in the asymmetric cooling scheme. Due to the differences in physics, once we switch from the asymmetric scheme to the symmetric one by applying an additional microwave, although we may have a sudden transition from to , cooling rate would significantly drop down unless other conditions, such as the appropriate laser strength, are also satisfied.
Under a realistic circumstance, the nNV center is influenced by a finite-temperature environment, where the phonon occupations in the vibrational degrees of freedom satisfy Bolzmann distribution as in units of . As such, an additional Lindblad operator should be involved in the Lindblad master equation with
Considering the vibrational decay rate with the quality factor prl-85-4458 (); prl-103-227203 (); oe-21-029695 (); pra-67-033402 (), the final average number of phonons with the cooling rate can be obtained by solving the rate equation prl-103-227203 (); HP-FP (). The term turns to be most important in determining when . Ideally, the final average phonon number is , which vanishes if .
Robustness. - A working scheme for cooling should be highly robust to the parameter fluctuation, which is essential to experimental implementation. In our case, the deviations from the conditions of Eqs. (3) and (5) yield second-order effects away from the ideal final average phonon numbers, i.e., and supplementary information (). In contrast, the EIT cooling and the Stark-shift-gate cooling are more sensitive to such deviations, whose effect is reflected in the first-order expansion. Further considering the inaccuracy of in our schemes, we find and supplementary information (), i.e., only the third-order effect in the cooling.
Experimental feasibility. - For a nNV center levitated in an optic trap pra-90-033834 (); prl-111-180403 (); pra-88-033614 (), the diameter of the nNV sphere is about 20 - 200 nm, the vibrational frequency is 500 kHz, and the magnetic field gradient T/m and the coupling strength kHz are available. Moreover, the mechanical quality factor Q is relevant to the pressure of the vacuum and can be as high as for a pressure Torr pnas-107-1005 (). The decay from is MHz and the environment can be at room temperature pra-88-033614 (); pnas-107-1005 () or cryogenic (such as 1 mK) pra-90-033834 (). For a specific calculation, we choose kHz and kHz, implying within the working region of the symmetric cooling scheme. Provided that the environment is at room temperature TK, and MHz, we may achieve after cooling for 100 s, better than the cooling by the Stark-shift-gate scheme (only reaching ).
With respect to the symmetric cooling scheme, the asymmetric scheme favors a smaller and meanwhile saves a microwave irradiation. For a nNV center attached at the end of a cantilever, since the vibrational frequency of the cantilever is usually of the order of few MHz or larger, the asymmetric scheme applies to such a case. Provided that the environmental temperature T mK, MHz, , kHz and MHz, we may cool the cantilever’s vibration by the asymmetric scheme down to after cooling for 90 s.
Discussion. - Two special characteristics of the NV center, as the difference from the atoms’, should be mentioned. First, there is a leaking channel out of the four-level configuration employed in Fig. 1(c, d), i.e., from to , and finally down to . For the three-level structure involving and , a pumping from back to is required in order to accomplish the desired cooling oe-21-029695 (). In contrast, there is no need to additionally consider such a pumping in the present schemes since is involved in the four levels. Second, the detrimental influence from nuclear spin bath of and should be seriously considered in the nNV center. This unexpected effect is neglected in above treatment, but actually leads to slight deviation from the dark-state condition and to inefficiency in removing the first-blue sideband transition. Simple estimate involving the nuclear spin noise can be found in the Supplementary Material supplementary information (). It is evident that this noise is beyond the scope of our schemes and should be suppressed by other approaches prl-102-057403 (); nl-12-2083 ().
Summary.- Our proposed efficient cooling schemes, based on quantum interference enhanced by the Stark-shift gates, can achieve a very low final average phonon number in some hot-topic systems with nNV centers. To the best of our knowledge, the schemes are the first proposal for cooling solid-state systems by completely eliminating heating effects from carrier transitions and the first-order blue sideband transitions. The two schemes favor slightly different conditions, and if employed appropriately, can be very general to achieve cooling for systems with four internal levels and a wide range of vibrational frequencies. Besides the two systems exemplified above, our cooling schemes can readily apply to other spin-vibration coupling systems involving four internal levels.
This work is supported by National Fundamental Research Program of China under Grants No. 2012CB922102 and No. 2013CB921803, and by National Natural Science Foundation of China under Grants No. 11274352 and No. 11304366.
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