Photon cooling by dispersive atom-field coupling with atomic postselection
PPhoton cooling by dispersive atom-field couplingwith atomic postselection
Felipe Oyarce and Miguel Orszag , Instituto de F´ısica, Pontificia Universidad Cat´olica de Chile, Casilla 306, Santiago,Chile Centro de ´Optica e Informaci´on Cu´antica, Universidad Mayor, Camino la Pir´amide5750, Huechuraba, Santiago, ChileE-mail: [email protected]
April 2019
Abstract.
We propose, in a Ramsey interferometer, to cool the cavity field to itsground state, starting from a thermal distribution by a dispersive atom-field couplingfollowed by an atomic postselection. We also analyze the effect of the cavity and atomiclosses. The proposed experiment can be realized with realistic parameters with highfidelity.
In the realm of cavity QED, a variety of experiments developed for generatingnonclassical states of light assume that the cavity field is initially in its vacuumstate [1–15]. Nevertheless, in most situations, the system is unavoidable coupled to theenvironment, that produces detrimental effects on the ideal realization of an experiment.In this way, assuming that the environment is in thermal equilibrium with the system,a cavity mode contains thermal photons on average that have to be removed at thebeginning of each experiment. One way to remove the residual thermal photons issending across the cavity a number of atoms initially prepared in the lower atomiclevel | g (cid:105) and tuned in resonance with the cavity mode [8, 11, 12]. However, in order toprepare the cavity field in its vacuum state, a more efficient technique to absorb thermalphotons is by the principle of the rapid adiabatic passage (RAP), in which the atom-fieldfrequency is swept [16, 17]. These two techniques employ a cooling sequence of atoms toreduce the effective field temperature by energy exchange between the cavity field andthe atoms.In this letter, we present a theorical scheme to cool-down an initial thermal field tothe pure vacuum state through an atomic postselection of a sequence of atoms interactingwith the field stored in a cavity. Similar works has been proposed for generating Fockstates superpositions of a cavity field [18], creating quantum vibrational states andcooling a nanomechanical oscillator by performing a postselective measurement [19, 20].In particular, we consider a dispersive atom-field coupling of each atom in the sequence,and thus there is no energy exchange between the atoms and the field, which is themain difference with previous schemes presented. In the present work, we find the a r X i v : . [ qu a n t - ph ] A p r hoton cooling by dispersive atom-field coupling with atomic postselection C prepared in an initial state ρ c . In box O , the atoms are initiallyprepared in the state | e (cid:105) with a given velocity. Then, each atom of the sequence crossesthree cavities: R , C and R . In each of the R and R zones, there is a semiclassicalinteraction atom between each atom and a classical microwave field. This interaction isessential to prepare a superposition of the states | e (cid:105) and | g (cid:105) in R , and manipulate theatomic state in R after the interaction with the field in C . At the end of the setup, inthe ionization zones D e and D g , the atomic level is postselected by detecting the atomin the state | e (cid:105) or | g (cid:105) .to the coupling, the effective interaction between the atoms and the field producesan energy shift to the atomic state. This energy shift leads to a phase shift on theatomic state which can be measured by a Ramsey interferometer setup as shown infigure 1 [2]. In the zone C , between the classical microwave zones R and R used forRamsey interferometry, we have a superconducting cavity with an initial field given by ρ c (0) = (cid:80) nn (cid:48) ρ nn (cid:48) | n (cid:105)(cid:104) n (cid:48) | written as an expansion using a Fock basis. The sequence of N three-level atoms is preselected in the state ρ a (0) = (cid:78) Nk =1 | e k (cid:105)(cid:104) e k | in box O and injectedinto the setup with a controlled velocity that allows us to assume that there is onlyone atom flying in the setup at a given time. Then, the initial state of the multipartitesystem (cavity - atoms) reads as ρ ca (0) = ρ c (0) ⊗ ρ a (0) . (1)The Ramsey interferometer is realized by applying two classical pulses, each onein zones R and R . The semiclassical interaction between each pulse and the levels | e k (cid:105) and | g k (cid:105) of the k th atom makes it possible to manipulate the atoms and create hoton cooling by dispersive atom-field coupling with atomic postselection | e k (cid:105) and | g k (cid:105) levels [21]. Particularly, considering an atomictransition frequency ω eg resonant with the pulse frequency with a pulse phase of φ = π/ τ k = ∆ L/v k which satisfies Ω R ∆ τ k = π/
2. The time evolutionof the atom, in each zone R and R , is an unitary transformation R π/ given by [22] R π/ = 1 √ (cid:32) (cid:33) (2)where ∆ L is the length of the R and R zones, v k is the velocity of the atom and Ω R is the Rabi frequency.On the other hand, in cavity C , we consider the atom as a two-level system thatinteracts with a quantized mode of a radiation field. In general, this situation is describedby the Jaynes-Cummings model H ( k ) = (cid:126) ω ie σ ( k ) z + (cid:126) ωa † a + (cid:126) g ( aσ ( k )+ + a † σ ( k ) − ) . (3)Here, the interaction involves only the levels | i k (cid:105) and | e k (cid:105) of each atom, and level | g k (cid:105) doesnot participate. Consequently, the Pauli operators are σ ( k ) − = | e k (cid:105)(cid:104) i k | , σ ( k )+ = | i k (cid:105)(cid:104) e k | , σ ( k ) z = | i k (cid:105)(cid:104) i k | − | e k (cid:105)(cid:104) e k | . The single mode of the quantized field is represented by thecreation and annihilation operators a † and a , respectively. As we mentioned above,we are interested in the dispersive regime, when the detuning between the cavity fieldfrequency ω and the atomic transition frequency ω ie is large as compared to the coupling g , i.e. δ = ω ie − ω (cid:29) g √ n . The effective dispersive Hamiltonian in this regime can bewritten as [21] H ( k ) eff = (cid:126) g δ a † aσ ( k ) − σ ( k )+ . (4)Equation (4) tells us that in both the field and the atom, the number of excitations isconserved, so there is no energy transfer between them. Moreover, during the interactiontime, the dispersive Hamiltonian produces a phase shift on | e k (cid:105) proportional to thephoton number.Explicitly, the time evolution operator after an interaction time τ k = L/v k is U ( k ) eff = exp (cid:16) − i H ( k ) eff τ k / (cid:126) (cid:17) = exp (cid:16) − i ϕ k a † aσ ( k ) − σ ( k )+ (cid:17) , (5)where the interaction time τ k is written in terms of the cavity length L and the velocity v k of the k th atom. Also, ϕ k = g τ k /δ is the phase shift of one photon.Now, following the same general procedure as in [18] for the generation ofFock states superpositions of the field, we can describe the complete evolution ofan individual atom that crossed the zones R , C and R by the evolution operator U ( k ) = R π/ U ( k ) eff R π/ . Hence, the total evolution operator of N successive atomsinteracting with the cavities after a time τ is U ( τ ) = U ( N ) ...U (1) . (6)The procedure is based on a postselection of the atomic levels in a target state | ψ t (cid:105) over the multipartite state of the whole system ( ρ ca ) after its evolution with the hoton cooling by dispersive atom-field coupling with atomic postselection U . We will see that the main task of our cooling protocol is to determine thedifferent values of ϕ k combined with a proper postselection.The postselection of the target state takes place in two ionization zones D e and D g ,where the level of the atoms is detected in | e (cid:105) or | g (cid:105) . Since the order is not important,we choose a symmetric target state of the form | ψ t (cid:105) = (cid:18) C N e N (cid:19) / (cid:88) p | m , ..., m N (cid:105) , (7)with the summation taken over all the possible combinations of N e atoms on the | e (cid:105) level and N − N e on the | g (cid:105) level. The normalization factor C N e N = N ! / [ N e !( N − N e )!] isthe number of combinations of N e atoms on the | e (cid:105) level in a set of N atoms. Therefore,the evolved cavity field state after the atomic postselection is ρ c ( τ ) = (cid:104) ψ t | U ( τ ) ρ ca (0) U † ( τ ) | ψ t (cid:105) . (8)After some straightforward calculations, the unnormalized field state is ρ c ( τ ) = C N e N (cid:88) nn (cid:48) ρ nn (cid:48) N (cid:89) k =1 e i2 ϕ k N ( n (cid:48) − n ) c N − N e n,k c N − N e n (cid:48) ,k d N e n,k d N e n (cid:48) ,k | n (cid:105)(cid:104) n (cid:48) | , (9)where the coefficients are c n,k = cos( ϕ k n/
2) and d n,k = sin( ϕ k n/ P post = C N e N (cid:88) n ρ nn N (cid:89) k =1 c N − N e ) n,k d N e n,k . (10)In the following, we show how to generate the vacuum state | (cid:105) of the cavityfield by an appropriate atomic postselection starting from a thermal state of the field ρ c = (cid:80) n ρ nn | n (cid:105)(cid:104) n | , where ρ nn = n nt / [(1 + n t ) n +1 ] and n t being the average photonnumber. As we can see from equation (9), we should have N e = 0 to keep the | (cid:105) state.Thus, the normalized state of the field for an initial thermal state after the postselectionof N atoms in the | g (cid:105) level ( N e = 0) is ρ f = (cid:80) n ρ nn (cid:81) Nk =1 cos ( ϕ k n/ | n (cid:105)(cid:104) n | (cid:80) n ρ nn (cid:81) Nk =1 cos ( ϕ k n/ , (11)with postselection probability P post = ∞ (cid:88) n =0 ρ nn N (cid:89) k =1 cos (cid:16) ϕ k n (cid:17) . (12)As shown in equation (11), the parameters ϕ k have to be adequate to ensure that theoscillatory function (cid:81) Nk =1 cos ( ϕ k n/
2) multiplying the projectors | n (cid:105)(cid:104) n | is close to zerofor all the photon numbers except n = 0. We propose a sequence of atoms where the k thatom crosses with ϕ k = π/ k − in order to eliminate the photon numbers n = (2 m − k .Once the process has finished the postselection probability is P post → ρ = 1 / (1 + n t ).Figure 2 shows the fidelity between the vacuum state and the final state of the cavity hoton cooling by dispersive atom-field coupling with atomic postselection N atoms in | g (cid:105) . For an initial state witha mean photon number n t = 100, the cooling process converges after the detection of asequence of about 10 atoms. N F i d e li t y n t = 5 n t = 20 n t = 100 Figure 2: Fidelity between the vacuum and the final state of the cavity field given byEq. (11) using a sequence of N atoms interacting with ϕ k = π/ k − .In figure 3, we illustrate the convergence of the sequence with ϕ k = π/ k − ,considering that all the atoms are postselected in the state | g (cid:105) . In the top panel(a), we plotted an initial thermal state with n t = 3 .
6, whereas in the bottom panel(b), we show the final field state after the postselection of 5 atoms in the state | g (cid:105) .As seen, the final state is the vacuum photon state | (cid:105) , where the Wigner function W [ ρ t ] ( α ) = π n t +1 e − | α | / (2 n t +1) becomes sharper than the initial thermal state.In a realistic scenario, the quantum system is coupled to the environment andsuffers from decoherence effects. Since we are cooling photons, we are fighting againstthe thermalization effect of the reservoir. To simulate this scenario, we consider thatthe cavity field is initially in thermal equilibrium with an average photon number n t ,and that the atom-field system evolves with the following master equation: dρ S dt = 1 i (cid:126) [ H eff , ρ S ] + (cid:88) i (cid:20) L i ρ S L † i −
12 ( L † i L i ρ S + ρ S L † i L i ) (cid:21) , (13)where H eff is the dispersive coupling Hamiltonian of equation (4) for each atom in thesequence and the L i Lindbland operators are (cid:112)
Γ(1 + n t ) σ − , √ Γ n t σ + , (cid:112) κ (1 + n t ) a and hoton cooling by dispersive atom-field coupling with atomic postselection n . . . . . P h o t o n . P r o b . D i s t . − . − . . . . x − . − . . . . y n . . . . . . P h o t o n . P r o b . D i s t . ( b ) − . − . . . . x − . − . . . . y Figure 3: Cooling process of the thermal field. In (a) we show the initial initial thermalstate ( n t = 3 . | g (cid:105) . In both figures the photon numberdistribution and Wigner function are shown. The vacuum photon state is generated witha probability of P post ≈ . hoton cooling by dispersive atom-field coupling with atomic postselection √ κn t a † .Typically, in microwave experiments with circular Rydberg atoms, the relaxationof these atoms is negligible, when compared to the cavity damping time T c . However,we did consider the lifetime reduction by a factor (1 + n t ), as well as the field losses,but neglected the losses during the transit time through the Ramsey zones. The masterequation for the period without atoms ( t ∼ µ s), is given by: dρ c dt = (cid:88) i (cid:20) L i ρ S L † i −
12 ( L † i L i ρ S + ρ S L † i L i ) (cid:21) , (14)where the Lindbland operators are (cid:112) κ (1 + n t ) a and √ κn t a † . In our calculations, weused a cavity damping time T c = 1 /κ = 130 ms [23]. Also, the atomic lifetime is T a = 1 / Γ = 30 ms for circular Rydberg atoms of rubidium with principal quantumnumbers 51 or 50 [22]. We consider the cavity tuned at a frequency ω/ π = 51 . δ/ π = 245 kHz. The vacuum Rabi frequency is g/ π = 49kHz. All of these parameters are consistent with real experimental realizations [14].The temperature of a thermal state can be determined by the relation n t =1 / [exp ( (cid:126) ω/k b T ) − n t = 3 . T = 10 K (figure3a). After the cooling process and considering the effect of the reservoir, the fidelitybetween the final state and vacuum goes down to 98.3% (figure 4). In this case, the finalstate has a 99.7% fidelity with respect to a thermal state with n t = 0 .
017 correspondingto a temperature of T = 0 . ∼ . ϕ k proposed in our atomic sequence tocool down the thermal photons in the cavity by atomic postselection can be achievedby controlling the velocity of each atom by laser techniques. However, our schemerequires that each atomic sample contains deterministically only one atom, and this ischallenging in real experiments. Most of the experiments prepare the atomic samplesby weak laser excitation resulting in a Poissonian distribution for the number of atomsin the sample. Despite of this, we assume in this work a deterministic preparation ofsingle atoms considering some schemes proposed to achieve the single-atom preparationof Rydberg atoms using the called dipole blockade effect [24, 25].In summary, in this work we suggest a protocol to cool-down a thermal field toits vacuum state using a typical cavity QED-setup. Here, a sequence of atoms is sentinteracting (one at a time) dispersively with the cavity field. After the interaction theatoms are postselected in its ground state, so our protocol is probabilistic. This means hoton cooling by dispersive atom-field coupling with atomic postselection n . . . . . . P h o t o n . P r o b . D i s t . − . − . . . . x − . − . . . . y Figure 4: Cooling process of figure 3 considering thermal effects of the environment ata temperature T = 10 K (mean photon number n t = 3 . n t = 0 . T = 0 . ϕ k = π/ k − which rapidly eliminate the nonzero photon components.The reduction of the number of atoms needed in the process is important when therelevant system is coupled to a thermal reservoir with T (cid:54) = 0, since the whole processtakes less time. We model this situation using the general master equation in equation(13), where we have considered atomic and field losses by taking real experimentalparameters. Finally, we conclude that even in the presence of decoherence our protocolcan be done with high fidelity. Acknowledgments
MO acknowledges the financial support of the project Fondecyt (1180175).
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