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The amplitude and phase control of light signal retrieved from the cavity-based quantum memory

N.G.Veselkova, I.V. Sokolov, St. Petersburg State University, Russia.

ABSTRACT Quantum memory for light allows one to map photon quantum states to matter in coherent and reversible manner, i.e. to write-in light pulses to long-lived matter degrees of freedom and then readout them back from matter to light with quantum states preserving. Quantum memory is an essential part of quantum information protocols such as quantum computing, quantum repeaters, quantum networks etc. To process light signals retrieved from the quantum memory an optical pulses mixing method can be used as, for example, in the case of homodyne method, in entangled states detectors, etc. Control of amplitude-phase characteristics of the signal plays a crucial role in schemes of optical mixing, see [1,3] (theory) и [4] (experiment). We propose a method to manipulate the readout process from resonator memory based on the Λ - scheme of atomic levels that allows one to extract the signal of desired amplitude temporal profile [2, 5] and controlled (permanent in ideal case) phase. This method gives the opportunity in principle to obtain a high efficiency of optical mixing. OFF - RESONANT RAMAN QUANTUM MEMORU Here we consider a quantum memory protocol based on the off-resonant Raman interaction [2,3], when the common detuning of the fields ∆ from the frequencies of atomic transitions is much greater than the rate of spontaneous decay of the upper level

. We consider a sparse ensemble of cold Λ–type atoms (Fig. 1) placed in a high-Q single-port cavity. It is assumed that the initial number of spin excitations is much less than the total number of atoms N in the ensemble, i.e. practically all atoms are in the ground state

during the entire interaction time.

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We will consider the readout stage (Fig. 2) on which a long-lived collective atomic excitation (spin wave) is converted by the control field acting on the transition - into the output signal .

Classical control field acts on the - transition and detuned from its frequency by the amount ∆, . The coupling of the atomic ensemble with the applied control field is characterized by the Rabi frequency , which is used for coherent control of the amplitude and phase of the output signal. Under the action of the control field in the atomic medium induced Raman transitions occur, as a result, photons are emitted at a frequency of . We will take as an initial approximation that the frequencies are adjusted so that coincides with the frequency of resonator without consideration the refractive index of the medium, and energy levels are taken without regard to stark shifts in a strong field. Initially, slow amplitudes of control and signal fields, spin amplitude and the electronic polarization will be determined relative to the appropriate frequencies. In the future, we will be introduced slow amplitudes taking into account the physical frequency shifts.

The assumption of a negligible populations of levels and allows us to linearize the system of equations for the field and atomic variables. The equations of Heisenberg - Langevin [1, 3] in the semiclassical approximation at zero input signal (readout) have the form

Omitted amplitude of noise sources match the vacuum fields and do not contribute to normal ordered averages which are included, for example, in the readout efficiency.

The detuning ∆ is considered much more relaxation parameters of the atomic transition and cavity field

, coupling parameters of the atom medium with quantized and reference fields

, as well as much larger spectral widths of all non-stationary amplitudes of the fields. Now we introduce physically justified slow amplitudes taking into account the frequency shift of the cavity mode by the amount

due to the refractive index of the medium and the shift of level

by the amount

due to the stark effect in a strong reference field:

. The basic equations for new variables take the form (tilde at amplitudes omitted):

here

is the coupling parameter that determines the states exchange frequency between the collective spin and quantized resonator field. There appeared decay rates of field mode and spin amplitude due to the excitation of an atomic level

in the wing of the broadened absorption lines in transitions

and

. OPERATION OF THE CONTROL FIELD Here we explore how to control the amplitude and phase of the coupling parameter (applying the method of approximate impedance matching) to readout the state of the collective spin in the output pulse of a predetermined shape and the controlled phase. An analytical expression for the amplitude and phase of the coupling parameter will be obtained there. In the system of equations for slow variables it is convenient to switch to dimensionless time,

, divided to decay time of the field energy in the cavity and to introduce the dimensionless amplitudes

(1) Here

, where

is the cooperativity parameter which plays a role of the effective optical thickness in the resonator model [1],

(

) is the dimensionless coupling parameter. Requiring the following behavior of the spin amplitude

where

is the point at which the minimum of function

on the interval

is achieved, and considering the amplitude

as the real function the real amplitude

and phase

of the coupling parameter can be founded,

, by the reversal of system (1):

(2)

(3) it should be put

The expression for the modulus of the spin amplitude which enters in formulas (2), (3) is found from the equation of excitations fluxes balance,

, and is explicitly determined by the given temporary profile

. The readout efficiency defined as the ratio of the number of output photons to the initial number of spin excitations can be expressed through the solution of system (1) by formula

The coupling parameter founded above is substituted in the system of equations (1) which is solved for required values of the parameters. As a result of the numerical solution of the system on the interval

with the initial conditions

we can find the time dependence of the amplitude and phase of the output signal and the dependence of the readout efficiency on the duration of the signal

RESULTS We apply the method of approximate impedance matching for the output signal with a normalized Gaussian temporal profile having a length in time

and width

at the level of

(Fig. 6)

where

is determined by the normalization condition

. The parameters are taken following:

MHz,

ns» 22,6.

Fig. 7. Amplitude of the coupling parameter

Graphs of the amplitude and phase of the coupling parameter for a given amplitude of the output signal are shown in Fig. 7, 8. Pulse area equal to

corresponds to that founded dependence

. The Fig. 9, 10 show the additive to the phase of the coupling parameter due to the stark effect,

, and the sum of phases

and

Fig. 8. Phase of the coupling hhh parameter

Fig. 9. The Stark`s additive to the phase of the coupling parameter

Fig. 10. The sum of phases

and

Plots for the phase and amplitude of the output signal which is found by numerically solving the system of equations (1) with the above dependencies and , and the readout efficiency as a function of signal duration are shown in Fig. 11-13. At ns» 22,6 value of the efficiency is .

Gggggg

Fig. 11. The phase of the output signal

Fig. 12. The amplitude of the output signal

, given impulse

and the coupling parameter amplitude

Fig. 13. The readout efficiency

Below are graphs of the amplitude and phase of the output signal and the readout efficiency obtained for a given pumping Gaussian shape to reveal the advantages of the impedance matching method (width at the level of is approximately equal to 1,2 for comparison with the work of [3] and MHz, which corresponds to the pulse area of the coupling parameter ). The Gaussian shape of the coupling parameter is used in the numerical solution of a system of equations analogous to (1) but without including the phase additive due to the stark effect in the variables. In this case when ns» 22,6 efficiency value is .

The phase of the output signal

The amplitude of the output signal

and pumping

The readout efficiency

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