Contrast and phase-shift of a trapped atom interferometer using a thermal ensemble with internal state labelling

We report a theoretical study of a double-well Ramsey interferometer using internal state labelling. We consider the use of a thermal ensemble of cold atoms rather than a Bose-Einstein condensate to minimize the effects of atomic interactions. To maintain a satisfactory level of coherence in this case, a high degree of symmetry is required between the two arms of the interferometer. Assuming that the splitting and recombination processes are adiabatic, we theoretically derive the phase-shift and the contrast of such an interferometer in the presence of gravity or an acceleration field. We also consider using a"shortcut to adiabaticity"protocol to speed up the splitting process and discuss how such a procedure affects the phase shift and contrast. We find that the two procedures lead to phase-shifts of the same form.


Introduction
Inertial sensors based on interferometry [1] with freely falling atoms have demonstrated excellent performance in the measurement of gravity [2], gravity gradients [3] and rotations [4]. Atom interferometry with trapped atoms is much less well developed although it offers some advantages: interrogation times are not limited by the atoms' flight from the interaction region and one can hope to reduce the overall size of the device using technologies such as atom chips [5][6][7]. These advantages motivated our recent proposal for a trapped atom interferometer using thermal atoms [8], a situation closely analogous to white light interferometry in optics [9]. In it we discussed the importance of maintaining a high degree of symmetry in the two interferometer arms.
In that design we discussed use of internal state labelling of non-condensed ultra-cold atoms [6], essentially a Ramsey interferometer with an adiabatic spatial separation of the internal states. An adiabatic procedure however, has the disadvantage of severely limiting the speed of the splitting: the separation must be slow compared to the trap oscillation period. Here we will consider another approach inspired by recent work on 'shortcuts to adiabaticity' (STA) [10,11] which allows one to effect the separation more rapidly [10,12,13]. This technique is already in use in some experiments to move the position [13] and change the frequencies [14] of a trap filled with a thermal gas or a Bose-Einstein condensate [15]. Although a STA protocol is rather complex, we find that the resulting phase shift and interferometer contrast are of the same intuitive form as in the adiabatic case.
In this paper we consider a protocol similar to the one described in [6,8], namely a Ramsey interferometer with spatial separation of the internal states. Such a configuration has the advantage of providing an independent control on the two arms of the interferometer [8], and allows the phase to be measured by atom counting rather than fringe fitting. We take into account the possible effect of gravity or acceleration, and describe the dynamics of the splitting and recombination process in two particular cases. In the first case, we assume that the splitting and recombination process is slow enough that adiabatic approximation holds [8]. In the second case, we assume purely harmonic trap and derive an optimal interferometric sequence based on the STA technique [10,12]. This paper is organised as follows: in section 2, we describe the basic principles of the interferometer protocol we consider. In section 3, we discuss the phase-shift and contrast in the case of adiabatic splitting and recombination. In section 4, we then consider the whole interferometric sequence as a dynamical problem, and show, in the case of harmonic potentials, that STA [10,[12][13][14] can be used to reduce the splitting and recombination time. We give an expression for the dynamical phase-shift of the interferometer, including the effects of the splitting and recombination ramps, the temperature and the asymmetry between the trapping potentials.

Interferometer protocol
In this section, we briefly recall the interferometer protocol described in [8], and that we will consider in the rest of this paper. Consider an ensemble of atoms with two levels | ñ a and | ñ b . A typical interferometric sequence starts with a p 2 pulse to put the atoms in a coherent superposition of | ñ a and | ñ b with equal weights ( figure 1(b)). Then the two internal states are spatially separated (the splitting period, figure 1(c)), held apart (the interrogation period, figure 1(d)) and recombined (the merging period, figure 1(e)) using state-dependent potentials ( )  V z t , i which are only seen by atoms in internal state | ñ i . We note  z the position operator, t the time and = i a b , . We suppose that the design of the interferometer [8] allows V a =V b at the beginning and at the end of the sequence. Finally, another p 2 pulse closes the interferometer (figure 1(f)). Between the two p 2 pulses, the system can be described by the following Hamiltonian [8]: where  p is the momentum operator and w ab is the energy difference between the two internal states at the beginning and at the end of the interferometric sequence. Before the first p 2 pulse (labelled by t = 0), we assume that the state of the atomic cloud is the same as in [8] (i.e. in the internal state | ñ a , at thermal equilibrium with temperature T in the trapping potential V a ). Thus we describe it by the same density matrix | ( ) | | ( )|  r = å ñ ñá á p n a a n 0 0 |( )  ñá H a a t and constitute an orthonormal basis (the same notation will be used for | |( )  ñá H b b t later on in this paper). As in [8] we neglect the effect of collisions in the atomic cloud during the interferometric sequence (i.e. we do not have damping term in the Liouville equation for the evolution of the density operator), thus, due to the choice of the | ( ) | ñ ñ n t i i , the p n stay constant during the interferometric sequence. The effect of a p 2 pulse is modelled by: Schematic of a Ramsey interferometer protocol using internal state labelling. Time increases from (a) to (f). The size of the blue (respectively orange) shaded-disc, represents the population in state | ñ a (respectively | ñ b ). The blue (respectively orange) arrows indicate the direction of the displacement of the trap ( ) are identical and not drawn superimposed for clarity. A typical sequence is as follows: (a) the atomic cloud is prepared in internal state | ñ a and (b) a first p 2-pulse puts the atoms in a coherent superposition of the two internal states | ñ are moved to spatially separate the two internal states, (d) then the two internal states are held apart, (e) and the two trapping potentials ( ) are moved to spatially recombine the two internal states. (f) A second p 2-pulse closes the interferometer.
where we have neglected the finite duration of the pulse, f is the phase of the electromagnetic field at the beginning of the pulse, and ω the frequency of the electromagnetic field. This model is valid in the case where d w w =ab is the detuning from the atomic resonance, and Ω is the Rabi frequency.
Just after the second p 2 pulse (labelled by = t t f , where t f is the time between the two pulses), and using the hypothesis ( ) , . As in [8], the physical quantity measured in this interferometer is the total population in each internal state. We choose to write the total population in | ñ a , leading, from equation (3), to: n n a f f where we introduce the contrast: and the phase-shift:

Phase-shift and contrast in the adiabatic case
In this section, we assume that the time variations of ( ) are slow enough that the adiabatic approximation can be applied, as discussed in [8]. A more general non-adiabatic case will be considered in section 4. We furthermore assume that the path in parameter space describing the changes in retraces itself, such that the geometrical phase factors vanish [16] and thus are the adiabatic eigen-energies of | |( )  ñá H i i t . Moreover, we assume for simplicity that the duration of the splitting and merging period are much smaller than the duration of the interrogation period, such that the effect of splitting and merging on the phase shift and contrast can be neglected (taking into account more realistic interferometric sequences, as described in [8], does not change the conclusions drawn in this section). We can thus write the phase accumulated by | ( ) | ñ ñ n t i is difference between the eigen energies of the two traps for the same vibrational level.

Rule of thumb for the coherence time
A very convenient rule of thumb to infer the coherence time can be derived from equation (5) by considering the second order Taylor expansion of C under the assumption | |  dw t 1 n . This leads to ( ) where t c is understood as the coherence time, with the following expression for t c : In other words, the inferred decoherence ratet c 1 is on the same order of magnitude as the standard deviation of the dw n , weighted by the Boltzmann factors p n .
If we furthermore assume that V a and V b correspond, during the interrogation period, to two harmonic traps with slightly different frequencies w a and w b , with | |  w w w a b ab , , equation (7) leads, in the case of a weakly degenerate gas  w kT a b , , to: It is obvious from equation (8) that t c increases with symmetry and decreases with temperature, as expected intuitively. This result differs from the exact calculation, in case of two harmonic potentials [8], only by a factor 3 . For a typical temperature of 500nK, equation (8) gives a symmetry-limited coherence time on the order of 15ms for a realistic value of the asymmetry  dw w 10 −3 [17]. In the case of non-harmonic traps, equations (5) or (7) can still be used with perturbatively-or numerically-estimated values of the eigen-energies.

Phase-shift in the presence of a gravity or acceleration field
In the rest of this paper, we consider the case where ( )  V z i is the sum of a harmonic potential and an acceleration or a gravity potential namely: where m is the atomic mass, w i are the trap frequencies, g is the acceleration or gravity field, z i is the trap centre (minimum of the trapping part of the potential) and w =z z g i i i cm 2 is the centre of mass position of the atoms. The phase difference ( ) j D t (equation (6)) after an interrogation time t, stemming from Hamiltonian (1) and potential (9), is given in this case by: where : In equation (10) ( ) j D t 0 arises from the spatial separation of the two internal states, and ( ) w w t ab describes the free evolution of the states. In equation (11), the first term is the classical difference in potential energy due to the presence of the acceleration or gravity field. The second term is an energy shift resulting from the addition of the harmonic potential with the linear  mgz term (see equation (9)). The third term is the difference of zero point energies of the two harmonic oscillators. The last term, which is temperature dependent, vanishes in two cases: (i) a symmetric interferometer (i.e. w w = a b ), (ii) zero-temperature. Equation (12) shows that not only the contrast depends on temperature (as was predicted in [8]) but also the phase-shift. We also predict a direct link between the phase-shift and the relative asymmetry of the two traps, as was previously pointed out in [18].

Beyond the adiabatic case: STA
Let us now consider the dynamical problem of splitting and recombination. As illustrated by the numbers given previously for the coherence time, it is not always possible to perform adiabatic splitting and recombination (which have to be longer than the trap period [8]), because the inverse of the inferred coherence time (15 ms) is on the same order of magnitude as usual trapping frequencies in atom chip experiments (typically between 10 Hz and 1 kHz [19]). We thus need to speed up the splitting and recombination. A solution could be the use of STA.

STA ramps
It has been demonstrated in [12,13] that non-trivial temporal ramps can be used to move an atomic cloud while keeping the population of the different quantum levels unchanged at the ends of the ramp, on the time scale of the trapping period (hence much faster than an adiabatic ramp [8]). Such ramp can also be used to move trapped cold ions [20][21][22] while keeping their initial state at the end of the ramp. We propose, in the following, to apply this technique, known as STA [10,[12][13][14], to the case of a trapped thermal atom interferometer. For simplicity, we only consider the case of a harmonic trap (for other potentials the reader is referred to [10] and references therein). We thus consider a trapping potential with a time-depend position and stiffness: Similar to the case of equation (9), we can rewrite these potentials as: To introduce the mathematical condition which must be fulfilled for the STA, we need to write a dynamical invariant ( ) can be found in the literature [12,24,25]. After adapting them to include the presence of g, we obtain: where w 0 is an arbitrary angular frequency and r i and z i cm are solutions of the following equations: Equation (16) is the Ermakov equation and equation (17) is the classical linear oscillator. Physically, z i cm is the centre of mass of the atomic cloud obeying equation (17), and r i is proportional to the cloud size [12]. For a given time = t t p , the populations of the different quantum levels will be the same as [10,12]. This imposes in particular the following conditions on r i and z i cm at t m p , : are provided by (16) and (17). Together with (18) they form the STA conditions at t m p , . In order to find a temporal ramp on w i and z i for the splitting, we need to solve equations (16), (17) and (18). To do this, as we have six conditions on r i and six on z i cm , we take a fifth-order polynomial ansatz for r i and z i cm [12][13][14]. The frequency ramp is first found from r i and (16) and the trap position z i is then deduced from w i , z i cm and (17). To give a numerical example, the following parameters are taken (times are defined in figure 2): , g is the gravitational acceleration and the maximum separation distance between the two internal states is 200μm. This example is shown in figure 2, where we use the same ramp for recombination and splitting. Numerically we were not able to find t 1 significantly lower than 2ms while preserving a smooth ramp for the frequency (without imaginary frequencies to keep the trapping behaviour of the potential) and for the trap position. This is in accordance with [26] where it is stated that the minimum time is on the order of p w 2 i .

Contrast and phase-shift with STA ramps
For purposes of interferometry, the contribution to the overall phase shift of the splitting and merging period has to be taken into account, all the more since their duration is not negligible compared to the typical value of the coherence time inferred previously. The framework of the dynamical invariant ( )  I t i [23] provides a tool to compute this overall phase shift between t=0 and = t t f (i.e. during the whole interferometric sequence). Reference [23] gives the following generic solution | | ñ ñ t i of the Schrödinger equation with a time-dependent hamiltonian | |( )  ñá H i i t [23]. Adapting the results of [12,27,28] to the case of the trapped interferometer considered in this paper, we obtain: