Charged oscillator quantum state generation with Rydberg atoms
We explore the possibility of engineering quantum states of a charged mechanical oscillator by coupling it to a stream of atoms in superpositions of high-lying Rydberg states. Our scheme relies on the driving of a two-phonon resonance within the oscillator by coupling it to an atomic two-photon transition. This approach effectuates a controllable open system dynamics on the oscillator that permits the dissipative creation of squeezed and other non-classical states which are central to applications such as sensing and metrology or for studies of fundamental questions concerning the boundary between classical and quantum mechanical descriptions of macroscopic objects. We show that these features are robust to thermal noise arising from a coupling of the oscillator with the environment. Finally, we assess the feasibility of the scheme finding that the required coupling strengths are challenging to achieve with current state-of-the-art technology.
The interface between different types of quantum systems has been the subject of much attention in the quest for complex quantum technologies Wallquist:09 (); Kolkowitz:12 (); Xiang:13 (); Ladd:10 (). In order to combine advantages of various platforms, such as long coherence time, strong interactions or low-loss transport Aspelmeyer:13 (), one has to be able to transfer quantum state between different systems. Alternatively, the interactions between two different quantum systems can be exploited to produce and probe quantum states Kuzmich:98 (); Lahaye:09 ().
Mechanical systems in particular have seen rapid experimental progress. Nowadays, micro- and nano-mechanical oscillators can be cooled down to the quantum regime, where the quantised dynamics of the oscillator motion and controlled interaction with other quantum systems have become possible Kleckner:08 (); Poot:12 (); Aspelmeyer:13 (); Peterson:16 (). An alternative to the typically used optomechanical interaction is to exploit electric forces to couple an atom to a charged oscillator Bariani:14 (); Rabl:09 (); Rabl:10 (); Seok:11 (); Sanz:15 (). The strong dipole moment of atoms excited to high principal number Rydberg states Saffman:09 (), allows strong free-space interaction between single atoms and a charged oscillator, without the need for a mediating cavity. Atomic dipole - oscillator dipole coupling allows single atom cooling and the construction of complex superpostions of phononic Fock states Law:96 (). Moreover, efficient coupling between Rydberg atoms and microwave cavities Hogan:12 (), acceleration of flying atoms Lancuba:14 () and creation of superpositions between different Rydberg states Facon:16 () all constitute well established technologies. At the same time, results in the fabrication of micromechanical oscillators with resonance frequencies matching Rydberg transitions in atomic systems, and with high quality factor are promising, particularly using single-crystal diamonds Tao:13 (); Burek:12 (). Additionally, these oscillators can be superconducting, and thus become chargeable on demand Bautze:14 ().
In this paper we exploit the coupling between flying Rydberg atoms and a charged mechanical oscillator. We show that when the oscillator is driven at two-phonon resonance and if the coupling between the atoms and the oscillator is sufficiently strong, the system dynamics results in a non-classical state of the oscillator, whose nature can be tuned by a suitable choice of the initial atomic state. The desired oscillator states are obtained after the passage of only tens of atoms corresponding to the initial transient period of an effective dissipative dynamics. Specifically we show that under the strong coupling condition one can create a squeezed or Schrödinger cat states of the oscillator which are robust with respect to realistic thermal noise. These states are particularly useful for fundamental tests of quantum physics and decoherence processes Bose:99 (); Arndt:14 (), quantum information and quantum simulation Lombardo:15 (), metrology and sensing of small forces Gilchrist:04 () or even for dark matter detection Bateman:15 () or to probe quantum gravity inspired models Ghobadi:14 (). While squeezed states of micromechanical oscillators have been produced Wollman:15 (); Pirkkalainen:15 (); Lecocq:15 (), the creation of large and robust Schrödinger cat states of macroscopic mechanical oscillators is yet to be achieved. We perform a feasibility study and find that with current state-of-the-art technology it is challenging to access the strong coupling regime. To ultimately reach the desired coupling strengths may necessitate further developments, such as the use of collectively enhanced coupling through oscillator arrays or atomic ensembles.
Ii The system
The system under consideration is shown in Fig. 1(a). It consists of a stream of single Rydberg atoms coupled to a charge at the tip of a micromechanical oscillator, which oscillates in the direction around the origin. We denote by the displacement operator of the oscillator, where and are the bosonic phonon creation and annihilation operators, the characteristic oscillator length, the effective mass of the oscillator and is the mechanical oscillation frequency. The atoms move along a path such that only one atom is interacting with the oscillator at a time.
ii.1 Single atom dynamics
In this article we consider two distinct situations: a single-phonon and two-phonon resonance (see Fig. 1). In the first case the atomic ground state , the excited state , is the transition frequency and the interaction is described by the interaction Hamiltonian , see Fig. 1(b). Here, is the atomic dipole of the transition and is the electric field at the position created by the oscillator charge. In the latter case, the two-phonon oscillator transition couples to a two-photon transition between Rydberg levels and , which are states with different principal quantum number, via an off-resonant manifold of states. We denote by the transition frequency and by the atom-oscillator detuning, which is assumed to be much larger than the energy separation of states within the manifold. The interaction Hamiltonian in this case reads , where () is the dipole moment of the () transition (see Appendix A).
ii.2 Single-phonon resonance
The first scenario we are studying is that of a single-phonon resonance, where . Under the assumption of small oscillator displacement as compared to the distance between the oscillator and the flying Rydberg atom, , where , one can expand the electric field in powers of . Using the rotating wave approximation, the interaction picture Hamiltonian reads (see Appendix A)
where is the time dependent coupling strength 111In principle, the form of the coupling between the atom and the oscillator depends on the geometry of a given mechanical mode and should be taken into account. For simplicity we assume that the atomic dipole couples to the center of mass motion of the mechanical oscillator as described in Appendix A.. For this resonant case, the time evolution can be solved exactly with the propagator , where
Here is the oscillator phonon occupation number, , is the integrated coupling strength and (2) is written in the basis. This is a situation corresponding to the micromaser physics as described for example in Ref. Englert:02 ().
The atoms are prepared identically and interact one at a time with the oscillator (see Fig. 1(a)) such that the evolution of the oscillator can be evaluated according to after the passage of each single atom. The initial state of each atom is assumed to be a superposition of the form
with the amplitude . The state of the oscillator can be determined at an arbitrary time iteratively as follows: the state of the oscillator after atoms have passed can be obtained by time-evolving the initial product state (where is the initial state (3) of the atom) with and subsequently tracing out the atomic degrees of freedom
The propagator gives the exact evolution of the system as an atom travels past. However, it is useful to describe the dynamics of the oscillator in terms of an approximate master equation. We derive the master equation in the limit where the change in the oscillator state due to the interaction with a single atom is small such that , where and is the rate by which the atoms fly by the oscillator. The master equation approach has the advantage that it provides useful insights in the dynamics of the system without explicit exact solution. It also allows for adding directly the coupling to a thermal bath Englert:02 (), as we shall discuss in detail in the case of the two-phonon resonance.
ii.3 Two-phonon resonance
In order to move beyond a displaced thermal state and achieve quantum states that are more complex we consider coherent two-phonon transitions of the oscillator that generate explicitly quantum effects. On the atomic side we consider two possible coupling mechanisms: direct single-photon and intermediate states mediated two-photon transition between a pair of atomic levels. As both situations lead to equivalent form of the effective interaction Hamiltonian, we first focus only on the latter which we analyze in detail. We then invoke the former in Sec. III for the sake of quantitative comparison.
The situation for atomic two-photon transition via an intermediate manifold of states coupled to a two-phonon transition of the oscillator is depicted in Fig.1 (c). When the intermediate manifold of states is detuned far enough from resonance with a single phonon it remains unpopulated and can be eliminated from the dynamics leaving an effective two-level system.
In the following we consider the case of a two-phonon resonance with the initial atomic state as described in Fig.1 (c). On two-phonon resonance () the intermediate levels are adiabatically eliminated and the interaction between the atom and the oscillator is described by the interaction picture Hamiltonian
As in the single-phonon resonance case, the time evolution of the system can be solved exactly using the propagator . Here
which is now written in the basis , and . Note that the evolution in the odd/even subspaces of the oscillator are independent of each other.
The two-phonon coupling between the atom and the oscillator is reminiscent of two-photon micromasers Gomes:99 (); Vidella:01 (); Ahmad:08 (), and we show here that it allows the creation of squeezed states, as suggested by the form of the Hamiltonian (6) Nunnenkamp:10 (). For the quantification of squeezing we introduce the standard quadrature observable
where . The quadrature angles correspond to the and quadratures, and the state is squeezed along if . The squeezing of mechanical motion was in fact achieved in recent experiments Wollman:15 (); Pirkkalainen:15 (); Lecocq:15 (). The manipulation of the oscillator state using Rydberg atoms at two-phonon resonance however goes beyond the squeezed state preparation and allows for creation of various other kinds of non-classical states. In order to quantify the non-classicality of the created states we use the negativity of the Wigner quasi-probability distribution , where is the spatial wavefunction of the oscillator Wigner:32 (). The negative volume of the Wigner function then reads Kenfack:04 ()
The exact evolution of the system can be solved by iteratively applying (4) where is replaced by and we take . The resulting state depends on the number of atoms that pass by the oscillator. The exact value of is not particularly important, as long as the number of atoms is sufficient to reach the desired non-classical state. For the following calculations, we fix , which fulfills this conditions for all considered states.
We now turn to numerical simulation of the exact evolution as described by eqs. (3),(4) and (7). The results of the simulation are summarized in Fig. 2. Fig. 2(a) shows the minimum variance of the state of the oscillator as a function of the integrated coupling strength and the atomic excited state population . The angle minimizing depends only on the relative phase between the atomic states (see Appendix C). For used in the simulation, . The negative volume of the Wigner function (9) is plotted in Fig. 2(b).
Finally, Fig. 2(c-f) show the Wigner function for specific values of and denoted by in Fig. 2(a). Points (c) and (e) show examples of squeezed states for small and large respectively. Points (d) and (f) show examples of states with significant negative regions of the Wigner function. The state shown in Fig. 2(d) has the qualitative features of a cat state Dodonov:74 (), which is of particular interest as it is used in metrology for small force sensing Gilchrist:04 () and in fundamental test of quantum mechanics Arndt:14 ().
Iii Thermal fluctuations and experimental considerations
We now investigate how robust the production of these quantum states is in the presence of thermal fluctuations. Combining the master equation for the interaction with the passing atoms, derived analogously to the single phonon case (see Appendix B), with the thermal processes gives
where the atomic part is
and the thermal part is
Here is the coupling of the oscillator to the thermal bath and is the mean phonon number of the bath at temperature .
To demonstrate how the coupling to the thermal bath deteriorates the oscillator quantum states, we solve the master equation (10) numerically for a total time corresponding to the passage of 30 atoms and initial thermal state with 222In practice, it might be difficult to control deterministically the exact number of the atoms interacting with the oscillator. We have verified, that varying the number of passing atoms from 25 to 35 results in only small changes (order of %) of . . In Fig. 3(a) we plot the negative volume of the Wigner function as a function of thermal coupling and mean thermal occupation number . The used parameters , correspond to the cat state in Fig. 2(d). Fig. 3(b) shows the minimum variance for parameters corresponding to the squeezed state in Fig. 2(c).
It follows from Fig. 2(a,b) and Fig. 3(a,b) that in order to create a non-classical state one requires and the atom passage rate should be maximized while minimizing and . With the help of specific examples, we demonstrate the performance of the scheme below.
Considering Hz and the state-of-the-art temperature mK corresponding to , we find for the cat state of Fig 3(a) that . Here denotes the value of for the system not coupled to a thermal bath ( for the parameters used in Fig. 3(a)). Similarly, using (10) with the parameters from Fig. 3(b), we find that for squeezing to be achieved one needs corresponding to mK.
Next, in order to assess what couplings can be achieved in a realistic experiment, we consider the following parameters: Cs Rydberg atoms with a transition between and which are separated by GHz Lorenzen:83 () corresponding to an oscillator resonant frequency GHz, which are achievable e.g. with clamped mechanical beams Huang_2004 () (although with smaller quality factors than assumed in this work) or single-crystal diamond nanobars Burek:12 (). The detuning between the oscillator frequency and the transition frequency is MHz, while the splitting MHz Lorenzen:83 (). We take the oscillator characteristic length m, the charge on the tip of the oscillator (compatible e.g. with aF capacitances of micron size electromechanical resonators operated with V voltages Yao_2000 (); LaHaye_2004 ()) and thermal bath coupling strength Hz (corresponding to a quality factor of the oscillator Burek:12 ()). For Rydberg states the atomic size is m, and the corresponding dipole moments are ( is the Bohr radius). For the atomic motion, we consider a simple linear trajectory with going from to , where we neglect any deflection of the atom’s path due to the interaction with the oscillator (the static monopole part of the field resulting from the charge Q can always be compensated by additional static charges; see Appendix A for details of the interaction). Assuming the atom-cantilever distance to be m we choose the atomic speed m/s and the rate of atoms atoms per second, giving the separation between successive atoms of m and the interaction time of couple of s. This guarantees, to a good approximation, that only one atom is interacting with the oscillator at a time and that one can neglect the decay of the Rydberg states which have lifetime of 100 s Feng:2009 (); Gallagher_1994 (). We then obtain for the integrated coupling strength .
We now turn our attention to the direct single-photon two-phonon resonance provided by the atomic dipole - oscillator quadrupole coupling as we show in Appendix A. Here, an analogous derivation leads to the integrated coupling strength which is smaller by orders of magnitude compared to in the two-photon two-phonon resonance scheme.
Iv Discussion and Outlook
We have explored a method of creating squeezed and non-classical states of a charged macroscopic mechanical oscillator. Such on-demand quantum state preparation constitutes a basic element of the mechanical oscillators state manipulation toolbox using atoms. Specifically, the squeezed and Schrödinger cat states that can be in principle generated might find applications as probes of decoherence processes of macroscopic bodies, in quantum information processing or in sensing and metrology. The values of the estimated couplings that are achievable with current state-of-the-art technology and typical parameter regimes turn out to be too small to be of a practical use. Further improvement might be sought e.g. by increasing the charge of the oscillator or by more suitable choice of the employed Rydberg states which would increase the dipole moment and decrease the two-photon detuning . Another possibility is to exploit the enhancement of the coupling when considering an ensemble of atoms coupled to an array of oscillators which we leave for future investigations.
Remark: After finishing our manuscript, we became aware of a related work Adbi:16 ().
J. M. would like to thank M. Marcuzzi and A. Armour for useful discussions. R. S. would like to thank A. Kouzelis for his comments. We thank W. Li for his help and the feedback on the manuscript. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 335266 (ESCQUMA). S.H. was supported by the German Research Foundation (DFG) through Emmy-Noether-grant HO 4787/1-1 and within SFB/TRR21.
- (1) M. Wallquist et al., Phys. Scr. T137, 014001 (2009).
- (2) S. Kolkowitz et al., Science 335, 1603 (2012).
- (3) Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Rev. Mod. Phys. 85, 623 (2013).
- (4) T. D. Ladd et al., Nature 464, 45 (2010).
- (5) M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 65 (2014).
- (6) A. Kuzmich, N. P. Bigelow, and L. Mandel, Europhys. Lett. 42, 481 (1998).
- (7) M. D. LaHaye et al., Nature 459, 960 (2009).
- (8) D. Kleckner et al., New J. Phys. 10, 095020 (2008).
- (9) M. Poot and H. S. van der Zant, Phys. Rep. 511, 273 (2012).
- (10) R. W. Peterson et al., Phys. Rev. Lett. 116, 063601 (2016).
- (11) F. Bariani, J. Otterbach, H. Tan, and P. Meystre, Phys. Rev. A 89, 011801 (2014).
- (12) P. Rabl et al., Phys. Rev. B 79, 041302 (2009).
- (13) P. Rabl et al., Nat. Phys. 6, 602 (2010).
- (14) H. Seok, S. Singh, S. Steinke, and P. Meystre, Phys. Rev. A 85, 033822 (2011).
- (15) A. Sanz-Mora, A. Eisfeld, S. Wüster, and J. M. Rost, (2015).
- (16) M. Saffman, T. G. Walker, and K. Molmer, Rev. Mod. Phys. 82, 2313 (2009).
- (17) C. Law and J. Eberly, Phys. Rev. Lett. 76, 1055 (1996).
- (18) S. D. Hogan et al., Phys. Rev. Lett. 108, 063004 (2012).
- (19) P. Lancuba and S. D. Hogan, Phys. Rev. A 90, 053420 (2014).
- (20) A. Facon et al., arXiv:1602.02488 (2016).
- (21) Y. Tao and C. Degen, Advanced Materials 25, 3962 (2013).
- (22) M. J. Burek et al., Nano letters 12, 6084 (2012).
- (23) T. Bautze et al., Carbon 72, 100 (2014).
- (24) S. Bose, K. Jacobs, and P. L. Knight, Phys. Rev. A 59, 3204 (1999).
- (25) M. Arndt and K. Hornberger, Nat. Phys. 10, 271 (2014).
- (26) D. Lombardo and J. Twamley, Sci. Rep. 5, 13884 (2015).
- (27) A. Gilchrist et al., J. Opt. B 6, S828 (2004).
- (28) J. Bateman et al., Scientific reports 5, (2015).
- (29) R. Ghobadi et al., Phys. Rev. Lett. 112, 080503 (2014).
- (30) E. E. Wollman et al., Science 349, 952 (2015).
- (31) J. M. Pirkkalainen et al., Phys. Rev. Lett. 115, 243601 (2015).
- (32) F. Lecocq et al., Phys. Rev. X 5, 041037 (2015).
- (33) In principle, the form of the coupling between the atom and the oscillator depends on the geometry of a given mechanical mode and should be taken into account. For simplicity we assume that the atomic dipole couples to the center of mass motion of the mechanical oscillator as described in Appendix A.
- (34) B.-G. Englert, (2002).
- (35) A. F. Gomes, J. A. Roversi, and A. Vidiella-Barranco, J. Mod. Opt. 46, 1421 (1999).
- (36) A. Vidiella-Barranco, Opt. and Spectr. 91, 369 (2001).
- (37) M. A. Ahmad and S. Qamar, Opt. Comm. 281, 1635 (2008).
- (38) A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, Phys. Rev. A 82, 021806 (2010).
- (39) E. Wigner, Phys. Rev. 40, 749 (1932).
- (40) A. Kenfack and K. Zyczkowski, J. Opt. B 6, 396 (2004).
- (41) V. V. Dodonov, I. Malkin, and V. Man’ko, Physica 72, 597 (1974).
- (42) In practice, it might be difficult to control deterministically the exact number of the atoms interacting with the oscillator. We have verified, that varying the number of passing atoms from 25 to 35 results in only small changes (order of %) of .
- (43) C. Lorenzen and K. Niemax, Phys. Scr. 27, 300 (1983).
- (44) X. M. H. Huang, Ph.D. thesis, California Institute of Technology, 2004.
- (45) J. J. Yao, Journal of micromechanics and microengineering 10, R9 (2000).
- (46) M. LaHaye, O. Buu, B. Camarota, and K. Schwab, Science 304, 74 (2004).
- (47) Z.-G. Feng et al., Journal of Physics B: Atomic, Molecular and Optical Physics 42, 145303 (2009).
- (48) Rydberg atoms, edited by T. F. Gallagher (Cambridge University Press, Cambridge, 1994).
- (49) M. Abdi et al., Phys. Rev. Lett. 116, 233604 (2016).
- (50) Quantum Mechanics, edited by A. Messiah (Dover Publications, New York, 1999).
- (51) Theoretical Atomic Physics, edited by H. Friedrich (Springer, Berlin, 2006).
- (52) I. Shavitt and L. T. Redmon, The Journal of Chemical Physics 73, 5711 (1980).
Appendix A Atom-oscillator interaction
The interaction Hamiltonian between the dipole moment of an atom at position and the electric field created by a charge at position can be expressed as a power series in
with and the last line introduces notation for the coupling strengths that are used in the following. The first term in eq. (15) corresponds to a Coulomb interaction, which can be cancelled by additional static charges with opposite sign (see also Fig. 1(a)) and thus we omit it in the following. The matrices depend on the specific atomic transitions that couple to the electric field of the oscillator. We compute the matrix elements using the standard angular momentum theory as Messiah_1999 (); Friedrich_2006 ()
where , is the electron angular momentum, the total angular momentum and the projection of the total angular momentum on the axis. The operators are given by the relations and . When expressed in the coordinate basis, they are simply rescaled spherical harmonics , . The dipole matrix elements are then obtained with the help of the relation
where , is the total spin and , are the Wigner and symbols respectively. Note that we have absorbed the radial part of the dipole transition elements into the dipole moment amplitude .
a.1 Single-phonon transition
For a single-phonon transition we consider resonant transitions the and manifolds of an atom within the same principal quantum number, as shown in figure 4(a). The transition matrices in the basis read
We calculate the coupling strengths for a position . We find that and that . Note that the last ratio is independent of . Fig. 4(b) shows the dipole coupling strengths as a function of the scaled coordinate , where the coupling strengths have been normalized to the maximum value of . Since we neglect . This simplifies the description so that one can use only two of the four levels and we choose and .
With this two-level system the atom-oscillator Hamiltonian can be written as , where and
where , and we have used . When the - transition of the atom is resonant with the one-phonon transition of the oscillator and the interaction picture Hamiltonian reads
where and contain only terms oscillating at or higher frequency, which can be neglected through the rotating wave approximation and we have introduced the single-phonon coupling strength .
a.2 Two-phonon resonance
Here we consider a situation where a two-photon atomic transition between different principal quantum number S-states and couples to a two-phonon oscillator transition. The two-phonon transition is mediated by an off-resonant coupling to the and manifolds. We denote by , () the () transition frequencies and by , the respective detunings. In what follows, we refer to both manifolds combined as to the manifold. Formally, the () transitions are described by a dipole moment operator () with magnitude () respectively. The atom-oscillator interaction is given by the sum , where is given by (15), and the transition matrices (), for the () transitions now read
To first order in , the total Hamiltonian in the atomic basis reads
with , and the atom-oscillator coupling strengths , and similarly for , where is replaced by . Taking the rotating wave approximation, the interaction picture Hamiltonian is
If for all single phonon coupling rates , we can adiabatically eliminate the manifold to get an effective two-level atom. Such situation occurs for different species and a range of principal quantum numbers. For instance, taking Cs, for , for and GHz (the example considered in the main text) yields MHz and MHz. In order to simplify the treatment, we thus replace the detunings in eq.(24) by . This also motivates the introduction of the effective transition frequency between the combined manifold and the state such that