# Storing and processing optical information with ultra-slow light in Bose-Einstein condensates

###### Abstract

We theoretically explore coherent information transfer between ultra-slow light pulses and Bose-Einstein condensates (BECs) and find that storing light pulses in BECs allows the coherent condensate dynamics to process optical information. We consider BECs of alkali atoms with a energy level configuration. In this configuration, one laser (the coupling field) can cause a pulse of a second pulsed laser (the probe field) to propagate with little attenuation (electromagnetically induced transparency) at a very slow group velocity ( m/s), and be spatially compressed to lengths smaller than the BEC. These pulses can be fully stopped and later revived by switching the coupling field off and on. Here we develop a formalism, applicable in both the weak and strong probe regimes, to analyze such experiments and establish several new results: (1) We show that the switching can be performed on time scales much faster than the adiabatic time scale for EIT, even in the strong probe regime. We also study the behavior of the system changes when this time scale is faster than the excited state lifetime. (2) Stopped light pulses write their phase and amplitude information onto spatially dependent atomic wavefunctions, resulting in coherent two-component BEC dynamics during long storage times. We investigate examples relevant to Rb-87 experimental parameters and see a variety of novel dynamics occur, including interference fringes, gentle breathing excitations, and two-component solitons, depending on the relative scattering lengths of the atomic states used and the probe to coupling intensity ratio. We find the dynamics when the levels and are used could be well suited to designing controlled processing of the information input on the probe. (3) Switching the coupling field on after the dynamics writes the evolved BEC wavefunctions density and phase features onto a revived probe pulse, which then propagates out. We establish equations linking the BEC wavefunction to the resulting output probe pulses in both the strong and weak probe regimes. We then identify sources of deviations from these equations due to absorption and distortion of the pulses. These deviations result in imperfect fidelity of the information transfer from the atoms to the light fields and we calculate this fidelity for Gaussian shaped features in the BEC wavefunctions. In the weak probe case, we find the fidelity is affected both by absorption of very small length scale features and absorption of features occupying regions near the condensate edge. We discuss how to optimize the fidelity using these considerations. In the strong probe case, we find that when the oscillator strengths for the two transitions are equal the fidelity is not strongly sensitive to the probe strength, while when they are unequal the fidelity is worse for stronger probes. Applications to distant communication between BECs, squeezed light generation and quantum information are anticipated.

###### pacs:

03.67.-a, 03.75.Fi, 42.50## I Introduction

In discussions of quantum information technology quantumInfo , it has been pointed out that the ability to coherently transfer information between “flying” and stationary qubits will be essential. Atomic samples are good candidates for quantum storage and processing due to their long coherence times and large, controllable interactions, while photons are the fastest and most robust way to transfer information. This implies that methods to transfer information between atoms and photons will be important to the development of this technology.

Recently the observation of ultra-slow light (USL) Nature1 ; OtherUSL , propagating at group velocities more than seven orders of magnitude below its vacuum speed (), and the subsequent stopping and storing of light pulses in atomic media Nature2 ; OtherStoppedLight has demonstrated a tool to possibly accomplish this quantumProcessing . The technique relies on the concept of electromagnetically induced transparency (EIT)EIT in three level -configuration atoms (see Fig. 1(a)). A coupling light field is used to control the propagation of a pulse of probe light . The probe propagates at a slow group velocity and, as it is doing so, coherently imprints it’s amplitude and phase on the coherence between two stable internal states of the atoms, labelled and (which are generally particular hyperfine and Zeeman sublevels). Switching the coupling field off stops the probe pulse and ramps it’s intensity to zero, freezing the probe’s coherent (that is, intensity and phase) information into the atomic media, where it can be stored for a controllable time. Switching the coupling field back on at a later time writes the information back onto a revived probe pulse, which then propagates out of the atom cloud and can be detected, for example, with a photo-multiplier tube (PMT) (see Fig. 1(b)). In the original experiment Nature2 , the revived output pulses were indistinguishable in width and amplitude from non-stored USL pulses, indicating the switching process preserved the information in the atomic medium with a high fidelity.

In addition to the ability to store coherent information, compressing ultra-slow probe pulses to lengths shorter than the atomic cloud puts the atoms in spatially dependent superpositions of and , offering novel possibilities to study two component BEC dynamics HauTalks . In SolitonVortex slow light pulses propagating in Bose-Einstein condensates (BECs) BEC of sodium were used to create density defects with length scales , near the BEC healing length, leading to quantum shock waves and the nucleation of vortices.

Motivated by both observations of coherent optical information storage in atom clouds and the interesting dynamics which slow and stopped light pulses can induce in BECs, the present paper theoretically examines new possibilities offered by stopping light in BECs. We first present a novel treatment of the switching process and establish that the switch-off and switch-on of the coupling field can occur without dissipating the coherent information, provided the length scale of variations in the atomic wavefunctions are sufficiently large. We find that the switching of the coupling field can be done arbitrarily fast in the ultra-slow group velocity limit . This is consistent with previous work Nature2 ; stoppedLight ; fastStopped ; stoppedTheory , but goes further as it applies to pulses in both the weak and strong probe regimes strongStopped . We also present the first explicit calculation of how the behavior of the system differs when one switches the coupling field faster than the natural lifetime of the excited state (see Fig. 1(a)) and find the information is successfully transferred even in this regime. Furthermore we will see the analysis presented here, being phrased in terms of the spatial characteristics of the atomic wavefunctions, is well suited to addressing the issue of storage for times long compared to the time scale for atomic dynamics.

Investigation of this very issue of longer storage times constitute the central results of the paper. In this regime, there are important differences between optical information storage in BECs versus atom clouds above the condensation temperature (thermal clouds), which have been used in experiments to date. During the storage time, the light fields are off and the atomic wavefunctions evolve due to kinetic energy, the external trapping potential, and atom-atom interactions. (We label all these external dynamics to distinguish them from couplings between the internal levels provided by the light fields.) To successfully regenerate the probe pulse by then switching on the coupling field, there must be a non-zero value of the ensemble average (over all atoms) of the coherence between and . When revivals were attempted after longer storage times in thermal clouds (several milliseconds in Nature2 ) the external dynamics had washed out the phase coherence between and (due to the atoms occupying a distribution of single particle energy levels) and no output pulses were observed. In zero temperature BECs the situation is completely different. There each atom evolves in an identical manner, described by a non-linear Schrodinger equation (Gross-Pitaevskii equation BEC ), preserving the ensemble average of the coherence during the external dynamics. That is, even as the amplitude and phase of the wavefunctions representing the BEC evolve, the relative phase of the two components continues to be well defined at all points in space. Thus, if we switch the coupling field back on, we expect that the evolved BEC wavefunctions will be written onto a revived probe field. In this way, the probe pulses can be processed by the BEC dynamics.

We show several examples, relevant to current Rb-87 BECs, of the interesting two component dynamics which can occur during the storage time. This atom possesses internal states with very small inelastic loss rates inelastic , and thus, long lifetimes multipleComponentJILA . We find that, depending upon the relative scattering lengths of the internal states involved and the relative intensity of the probe and coupling fields, one could observe the formation of interference fringes, gentle breathing motion, or the formation and motion of two-component (vector) solitons twoCompSolitons . In particular we find that using the levels and in Rb-87 could allow for long, robust storage of information and controllable processing. In each case, we observe the amplitude and phase due to the dynamical evolution is written onto revived probe pulses. These pulses then propagate out of the BEC as slow light pulses, at which point they can be detected, leaving behind a BEC purely in its original state .

For practical applications of this technique to storage and processing, one must understand in detail how, and with what fidelity, the information contained in the atomic coherence is transferred and output on the light fields. Thus, the last part of the paper is devoted to finding the exact relationship between the BEC wavefunctions before the switch-on and the observed output probe pulses. We find equations linking the two in the ideal limit (without absorption or distortion). Then, using our earlier treatment of the switching process, we identify several sources of imperfections and calculate the fidelity of the information transfer for Gaussian shaped wavefunctions of various amplitudes and lengths. In the weak probe limit, we find a simple relationship between the wavefunction in and the output probe field. We find that optimizing the fidelity involves balancing considerations related to, on one hand, absorption of small length scale features in the BEC wavefunctions and, on the other hand, imperfect writing of wavefunctions which are too near the condensate edge. For stronger probes, we find a more complicated relationship, though we see one can still reconstruct the amplitude and phase of the wavefunction in using only the output probe. In this regime, we find that the fidelity of the writing and output is nearly independent of the probe strength when the oscillator strengths for the two transitions involved ( and ) are equal. However, unequal oscillator strengths lead to additional distortions and phase shifts, and therefore lower fidelity of information transfer, for stronger probes.

We first, in Section II, introduce a formalism combining Maxwell-Schrodinger and Gross-Pitaevskii equations BEC , which self-consistently describe both the atomic (internal and external) dynamics as well as light field propagation. We have written a code which implements this formalism numerically. Section III presents our novel analysis of the switching process, whereby the coupling field is rapidly turned off or on, and extends previous treatments of the fast switching regime, as described above. Section IV shows examples of the very rich variety of two-component dynamics which occur when one stops light pulses in BECs and waits for much times longer than the characteristic time scale for atomic dynamics. Section V contains our quantitative analysis of the fidelity with which the BEC wavefunctions are transferred onto the probe field. We conclude in Section VI and anticipate how the method studied here could eventually be applied to transfer of information between distant BECs, and the generation of light with squeezed statistics.

## Ii Description of ultra-slow light in Bose-Einstein condensates

We first introduce our formalism to describe the system and review USL propagation within this formalism. Assume a Bose-condensed sample of alkali atoms, with each atom in the BEC containing three internal (electronic) states in a -configuration (Fig. 1(a)). The states and are stable and the excited level , radiatively decays at . In the alkalis which we consider, these internal levels correspond to particular hyperfine and Zeeman sublevels for the valence electron. When the atoms are prepared in a particular Zeeman sub-level and the proper light polarizations and frequencies are used Nature2 ; SolitonVortex , this three level analysis is a good description in practice. All atoms are initially condensed in and the entire BEC is illuminated with a coupling field, resonant with transition and propagating in the direction (see Fig. 1(b)). A pulse of probe field, with temporal half-width , resonant with the transition, and also propagating in the direction, is then injected into the medium. The presence of the coupling field completely alters the optical properties of the atoms as seen by the probe. What would otherwise be an opaque medium (typical optical densities of BECs are ) is rendered transparent via electromagnetically induced transparency (EIT) EIT , and the light pulse propagates at ultra-slow group velocities ( m/s)Nature1 . As this occurs, all but a tiny fraction of the probe energy is temporarily put into the coupling field, leading to a compression of the probe pulse to a length smaller than the atomic medium itself Nature1 ; Nature2 ; HarrisHau .

To describe the system theoretically, we represent the probe and coupling electric fields with their Rabi-frequencies , where are slowly varying envelopes of the electric fields (both of which can be time and space-dependent) and are the electric dipole moments of the transitions. The BEC is described with a two-component spinor wavefunction representing the mean field of the atomic field operator for states and . We ignore quantum fluctuations of these quantities, which is valid when the temperature is substantially below the BEC transition temperature FiniteTemp . The excited level can be adiabatically eliminated adiabatic when the variations of the light fields’ envelopes are slow compared to the excited state lifetime (which is 16 ns in sodium). The procedure is outlined in the Appendix. The functions evolve via two coupled Gross-Pitaevskii (GP) equations BEC . For the present paper, we will only consider dynamics in the dimension, giving the BEC some cross-sectional area in the transverse dimensions over which all dynamical quantities are assumed to be homogeneous. This model is sufficient to demonstrate the essential effects here. We have considered effects due to the transverse dimensions, but a full exploration of these issue is beyond our present scope. The GP equations are SolitonVortex ; thesis :

(1) | |||||

where is the mass of the atoms and we will consider a harmonic external trapping potential . The potential can in general differ from . For example, in a magnetic trap the states and can have different magnetic dipole moments. However, in the examples we consider the potentials are equal to a good approximation, and so is assumed in the calculations. Atom-atom interactions are characterized by the , where is the total number of condensate atoms and are the s-wave scattering lengths for binary collisions between atoms in internal states and . The last pair of terms in each equation represent the coupling via the light fields and give rise to both coherent exchange between as well as absorption into . In our model, atoms which populate and then spontaneously emit are assumed to be lost from the condensate (Fig. 1(a)), which is why the light coupling terms are non-Hermitian.

The light fields’ propagation can be described by Maxwell’s equations. Assuming slowly varying envelopes in time and space (compared to optical frequencies and wavelengths, respectively) and with the polarization densities written in terms of the BEC wavefunctions these equations are SolitonVortex ; thesis :

(2) |

where are the dimensionless oscillator strengths of the transitions and is the resonant cross section ( and for the lines of sodium and Rb-87, respectively). These equations ignore quantum fluctuations of the light fields, just as the GP equations (II) ignore quantum fluctuations of the atomic fields. Note that in the absence of any coherence these equations predict absorption of each field with the usual two-level atom resonant absorption coefficient (the atom density times the single atom cross-section).

For much of our analysis, we will solve Eqs. (II)-(II) self consistently with a numerical code, described in SolitonVortex ; thesis ; numerics . In SolitonVortex , this was successfully applied to predict and analyze the experimental observation of the nucleation of solitons and vortices via the light roadblock. When the light fields are off, the two states do not exchange population amplitude and Eqs. (II) reduce to coupled Gross-Pitaevskii equations for a two-component condensate. By contrast, when they are on and the probe pulse length (and inverse Rabi frequencies ) are much faster than the time scale for atomic dynamics (typically ms), the internal couplings induced by the light field couplings will dominate the external dynamics.

For our our initial conditions, we consider atoms are initially in in the condensed ground state. We determine the wavefunction of this state numerically by propagating (II) in imaginary time imaginaryTime , though the Thomas-Fermi approximation TF provides a good analytic approximation to . In the cases presented in this paper, we choose the transverse area so that the central density and chemical potential are in accordance with their values in a trap with transverse frequencies , as in previous experiments SolitonVortex . In Fig. 2 we consider an example with sodium atoms (with nm scatteringLengthNa ), and . The ground state density profile is indicated with the dotted curve in Fig. 2(b). In this case the chemical potential kHz.

With our initial ground state determined, consider that we initially () input a constant coupling field with a Rabi frequency and then inject a Gaussian shaped probe pulse at (see Fig. 1(b)) with a temporal half-width and a peak Rabi frequency . We define our times such that corresponds to the time the peak of the pulse is input. The dotted and dashed curves in Fig. 2(a) show, respectively, a constant coupling input MHz and a weaker input probe pulse with peak amplitude MHz and width s.

Solving Eqs. (II)-(II) reveals that, when the conditions necessary for EIT hold, the pulse will compress upon entering the BEC and propagate with a slow group velocity. As it does this, it transfers the atoms into superpositions of and such that thesis :

(3) |

which is a generalization to BECs of the single atom dark state darkState . Thus, the probe field imprints its (time and space dependent) phase and intensity pattern in the BEC wavefunctions as it propagates. When (3) is exactly satisfied, it is easily seen that the two light field coupling terms in each of Eqs. (II) cancel, meaning that and are completely decoupled from the excited state . However, time dependence of causes small deviations from (3) to occur, giving rise to some light-atom interaction which is, in fact, the origin of the slow light propagation. If one assumes the weak probe limit () and disregards terms of order the group velocity of the probe pulse is HarrisHau ; thesis :

(4) |

and so is proportional the coupling intensity and inversely proportional to the atomic density . The half-width length of the pulse in the medium is , which is a reduction from it’s free space value by a factor , while its peak amplitude does not change. Thus only a tiny fraction of the input probe pulse energy is in the probe field while it is propagating in the BEC. Most of the remaining energy coherently flows into the coupling field and exits the BEC, as can be seen in the small hump in the output coupling intensity (thin solid curve in Fig. 2(a)) during the probe input. This has been dubbed an adiabaton adiabatons . The magnitude of the adiabaton is determined by the intensity of the probe pulse.

In the weak probe limit never significantly deviates from the ground-state wavefunction and the coupling field is nearly unaffected by the propagation (both of these results hold to ). In this case (3) shows follows the probe field as the pulse propagates. The arrow in Fig. 2(a) indicates a time where the probe has been completely input and has not yet begun to output. During this time, the probe is fully compressed in the BEC and Fig. 2(b) shows the atomic densities in and at this time. The spatial region with a non-zero density in corresponds to the region occupied by the probe pulse (in accordance with (3)). Equation (3) applies to the phases as well as the amplitudes, however the phases in this example are homogenous and not plotted.

Once the pulse has propagated through the BEC, it begins to exit the side. The energy coherently flows back from the coupling to probe field, and we see the output probe pulse (thick solid curve). Correspondingly, we see a dip in the coupling output at this time. In the experiments the delay between the input and output probe pulses seen in Fig. 2(a) is measured with a PMT (Fig. 1(b)). This delay and the length of the atomic cloud is used in to calculate the group velocity. The group velocity at the center of the BEC in the case plotted is m/s.

Note that the output pulse plotted in Fig. 2(a) at is slightly attenuated. This reduction in transmission is due to the EIT bandwidth. The degree to which the adiabatic requirement is not satisfied, will determine the deviation the wavefunctions from the dark state (3), which leads to absorption into and subsequent spontaneous emission. Quantitatively, this reduces the probe transmission (the time integrated output energy relative to the input energy) to thesis :

(5) |

where is the optical density. In the Fig. 2 example, . The peak intensity of the pulse is reduced by a factor while the temporal width is increased by (this spreading can be seen in Fig. 2(a)). The appearance of the large optical density in (5) represents the cumulative effect of the pulse seeing a large number of atoms as it passes through the BEC. To prevent severe attenuation and spreading, we see we must use , which is s in our example.

## Iii Fast switching and storage of coherent optical information

We now turn our attention to the question of stopping, storing and reviving probe pulses. We will show here that once the probe is contained in the BEC, the coupling field can be switched off and on faster than the EIT adiabatic time scale without causing absorptions or dissipation of the information Nature2 ; thesis . While we find no requirement on the time scale for the switching, we will obtain criteria on the length scales of which must be maintained to avoid absorption events. While previous work Nature2 ; fastStopped ; stoppedTheory had addressed the fast switching case in the weak-probe limit, here we obtain results that are valid even when strongStopped . As we will see, for shorter storage times where the external atomic dynamics play no role, these length scale requirements can be related to the adiabatic requirements on the input probe pulse width . The advantage of the current analysis is that it is easily applied to analyzing probe revivals after the atomic dynamics have completely altered the wavefunctions . In such a case it is these altered wavefunctions, rather than the input pulse width , which is relevant.

### iii.1 Analyzing fast switching in the dark/absorbed basis

Consider that we switch the coupling field input off at some time with a fast time scale . Fig. 2(c) plots the probe and coupling intensities as a function of at various times during a switch-off with . We see the probe intensity smoothly ramps down with the coupling field such that their ratio remains everywhere constant in time. Remarkably, for reasons we discuss below, the wavefunctions are completely unaffected by this switching process. Motivated by this, we will, in the following, assume that do not vary in time during this fast switching period, and later check this assumption.

To understand this behavior, it is useful to go into a dark/absorbed basis for the light fields, similar to that used in normalModes , by defining thesis :

(6) |

where . From (6) one sees that when the condensate is in the dark state (3), . Using the notation and transforming the propagation equations (II) according to (6), one gets:

(7) | |||||

(8) |

where

(9) |

and we have ignored the vacuum propagation terms in Eq. (II). They are unimportant so long as the fastest time scale in the problem is slow compared to time it takes a photon travelling at to cross the condensate (about ps). Note that represents the usual absorption coefficient weighted according to the atomic density in each of and . The terms arise from spatial variations in the wavefunctions, which make the transformation (6) space-dependent. The term represents an additional effect present when the light-atom coupling coefficient differs on the two transitions ( and is discussed in detail in Section V.3.

Consider for the moment a case with (implying ) and assume a region in which are homogenous (so ). Equation (8) shows that the absorbed field attenuates with a length scale , the same as that for resonant light in a two-level atomic medium, and less than m at the cloud center for the parameters here. One would then get after propagating several of these lengths. Conversely, (7) shows the dark light field experiences no interaction with the BEC and propagates without attenuation or delay.

However, spatial dependence in gives rise to in (7)-(8), introducing some coupling between and , with the degree of coupling governed by the spatial derivatives . A simple and relevant example to consider is the case of a weak ultra-slow probe pulse input and contained in a BEC, as discussed above (see Fig. 2(b)). The wavefunction has a homogenous phase and an amplitude which follows the pulse shape according to (3), meaning that scales as the inverse of the pulse’s spatial length .

It is important to note that this coupling is determined by spatial variations in the relative amplitude and does not get any contribution from variations in the total atomic density . To see this we note that if we can write the wavefunctions as , where are constants independent of , then and as defined in (III.1) vanish.

When variations in the relative amplitude are present, examination of (8) reveals that when the damping is much stronger than the coupling (), one can ignore the spatial derivative term in analogy to an adiabatic elimination procedure. When we do this, (8) can be approximated by:

(10) |

Strictly speaking, one can only apply this procedure in the region where has propagated more than one absorption length (that is for where is defined by ). However, is in practice only a short distance into the BEC (marked with arrows in Figs. 2(b)-(d)). When the probe has already been input, as in Fig.2(b), and therefore (see (6)) are already trivially zero for so (10) holds everywhere. We next plug Eq. (10) into Eq. (7), giving:

(11) |

These last two equations rely only on assumptions about the spatial derivatives of and not on the time scale of the switch-off (though we have assumed in adiabatically eliminating from our original Eqs. (II)-(II)).

These results allow us to conclude two important things which hold whenever and the probe has been completely input. First, the coefficients governing the propagation of in (11) are extremely small. The length scales are already generally comparable to the total BEC size and the length scales for changes in , given by (11), scale as these terms multiplied by the large ratio . (The term in (11) can lead to additional phase shifts, which we discuss in Section V.3). Therefore the dark field propagates with very little attenuation. As we have noted (and later justify and discuss in more detail) the wavefunctions (and therefore the propagation constant in brackets in (11)) are virtually unchanged during the switch-off. Under this condition, changes initiated in at the entering edge quickly propagate across the entire BEC. To apply this observation to the switch-off, we note when the pulse is contained so at the entering edge, (6) shows there. Switching off the coupling field at then amounts to switching off at and (11) shows that this switch-off propagates through the entire BEC with little attenuation or delay. Second, from (10) we see that as is reduced to zero is reduced such that the ratio remains constant in time.

A numerical simulation corroborating this behavior is plotted in Fig. 2(d). The dark field intensity is seen to switch-off everywhere as the coupling field input is reduced to zero over a s timescale, confirming that the changes in the input propagate across the BEC quickly and with little attenutation. The ratio is everywhere much smaller than unity and constant in time. The only exception to this is in a small region at the cloud entrance, where has not yet been fully damped. The plot of during the switch-off indeed demonstrates how (10) is a generally good approximation, but breaks down in this region. In the case plotted (and any case where the probe is fully contained) the wavefunction is so negligible in this region that is rather small and unimportant (note the scale on the right hand side of the plot).

Translating this back into the basis, we note from (6) that keeping means the probe must (at all ) constantly adjust to the coupling field via:

(12) |

Thus we see that smoothly ramps down with even if is ramped down quickly, as seen in Fig. 2(c).

Using our results for the light fields, we can now see why the wavefunctions do not change during the switching. Physically, the probe is in fact adjusting to maintain the dark state (12), and in doing so induces some transitions between and . However, only a fraction of the input energy is in the probe while it is contained in the medium, and so the probe is completely depleted before any significant change occurs in Nature2 . In fact, the energy content of the probe field right before the switch-off is less than 1/100th of a free-space photon in the case here. Thus, is completed depleted after only a fraction of one transition. Note that (12) is equivalent to (3). However, writing it in this way emphasizes that, during the switching, the probe is being driven by a reservoir consisting of the coupling field and atoms and adjusts to establish the dark state. This is contrast to the situation during the probe input, when many photons from both fields are being input at a specific amplitude ratio, forcing the atomic fields to adjust to establish the appropriate dark state. Plugging in our results for the light fields (10)-(11) into our Eqs. (II) we can calculate the changes that occur in and during the switch-off. Doing this, we find relative changes in are both smaller than , which can be made arbitrarily small for fast switching . The little change which does occur is due not to any process associated with the switching itself but is due to the small amount of propagation during the switch-off. It is therefore safe to assume (as we saw numerically) that are constant in time during the switch-off. One can also show with this analysis that the ratio of coherent exchange events (from to or vice-versa) to absorptive events (transitions to followed by spontaneous emission) is implying that the switch-off occurs primarily via coherent exchanges.

In the original stopped light experiment Nature2 , the coupling field was then switched back on to , after some controllable storage time , at a time . In that experiment, revivals were observed for storage times too short for significant external atomic dynamics to occur, so the state of the atoms was virtually identical at and . Then the analysis of the switch-on is identical to the switch-off as it is just the same coherent process in reverse. The probe is then restored to the same intensity and phase profile as before the switch-off. An example of such a case is plotted in Fig. 2(e). The output revived pulse then looks exactly like the normal USL pulse (compare the output in Figs. 2(a),(e)), as was the case in the experiment.

We have established the requirement is necessary and sufficient for coherent switching to occur. Under what conditions is this satisfied? When the switch-off occurs while the probe is compressed, it is always satisfied because of bandwidth considerations mentioned above (see Eq. (5)). Specifically, the input pulse must satisfy . However, this leads to a pulse width in the medium of , implying that is satisfied (as ). Therefore, any pulse which can successfully propagate to the cloud center, can be abruptly stopped and coherently depleted by a rapid switch-off of the coupling field. Of course, if the switch-on is then done before significant atomic dynamics, the same reasoning applies then. In this case, our requirements on the spatial derivatives are already encapsulated by the adiabatic requirements on the probe pulse. It is when the external atomic dynamics during the storage significantly change , that the analysis of the switch-on becomes more complicated. This is the central purpose of the Sections IV-V below.

As hinted above, when the pulse is not yet completely input, the switching can cause absorptions and dissipation of the information. In this case is significant in the region and so is significant for several absorption lengths into the BEC. Physically, the coupling field sees atoms in a superposition of and immediately upon entering the condensate rather than only atoms. Numerical simulations of this situation confirm this. During both the switch-on and switch-off, a significant number of atoms are lost from both condensate components, primarily concentrated in the region, and the revived probe pulse is significantly attenuated relative to the pulse before the storage.

Incidentally, this explains the apparent asymmetry between the probe and coupling in a stopped light experiment. Whenever both fields are being input, the temporal variations in both fields must be slow compared with the EIT adiabatic time scale, as the atomic fields must adjust their amplitudes to prevent absorptions. However, when one of the fields is no longer being input (like a contained probe pulse), the other field’s input can be quickly varied in time.

### iii.2 Switching faster than

When the switching is done quickly compared to the natural lifetime of (), the assumptions needed for adiabatic elimination of are no longer valid. The analysis of the switching then becomes more complicated but, remarkably, we find the quality of the storage is not reduced in this regime.

To see this, we performed numerical simulations of switching in this regime by solving Eqs. (25)-(A) (which are the analogues of (II)-(II) without the adiabatic elimination of ) for the same parameters as Fig. 2 but varying from to numerics3Comp . An example with is shown in Fig. 3. We plot the probe and coupling amplitudes versus time at a point near the center of the BEC, and zoom in on the regions near the switch-on and -off. Note that that while the coupling field smoothly varies to it’s new value in each case, the probe field slightly overshoots the value given by (12) before returning to it. In all cases with the probe amplitude experienced damped oscillations before reaching its final value. The frequency of the oscillations was determined by and the damping rate was .

During these oscillations, the absorbing field is in fact quite significant. However, the maximum value of and the total atomic loss due to spontaneous emissions events was completely independent of over the range we explored. To understand this, we note that even though is larger, the frequency components of the quickly varying coupling field are spread over a range much wider than . Thus a very small fraction of the energy in the field is near resonant with the atomic transition and it passes through the BEC unattenuated. Analyzing the problem in Fourier space one can see that the increase in for smaller is precisely offset by this frequency width effect, leading us to the conclusion that spontaneous emission loss is independent of .

Unlike the case of switching slowly compared with , the adjustment of the probe is not a coherent process. Rather, spontaneous emission damps the system until the dark state (12) is reached. However, because we are in the regime the amount of energy in the probe is much less than one photon, and thus still has virtually no impact on . This will not necessarily hold when and in stoppedTheory it was predicted that there are, in fact, adiabatic requirements for in this regime. However, in all cases of interest here this inequality is well satisfied and there is no restriction on .

## Iv BEC dynamics and processing optical information

We now turn to the question of the atomic dynamics during the storage time. These dynamics depend strongly on the relative scattering lengths of the states used, and the probe to coupling intensity ratio. We will see that the these dynamics are written onto revived probe pulses by switching the coupling field back on. The length scale requirement on variations in the relative amplitude derived above () plays a central role in the fidelity with which these dynamics are written. We will demonstrate with several examples, relevant to current Rb-87 experimental parameters.

### iv.1 Writing from wavefunctions onto light fields

In Nature2 the revived pulses were seen to be attenuated for longer storage times with a time constant of ms. The relative phase between the and (or coherence), averaged over all the atoms, washed out because the atoms, although at a cold temperature of , were above the Bose-condensation temperature and so occupied a distribution of energy levels. By contrast, a zero temperature two-component BEC will maintain a well defined phase (the phase between and ) at all , even as evolve. They evolve according to the two-component coupled GP equations (II) with the light fields set to zero (), and the initial conditions are determined by the superposition created by the input pulse at (see Fig. 2(b)). After such an evolution, we can then switch the coupling field back on and the probe will be revived according to (12) but with the new wavefunctions , evaluated at .

Here we examine a wide variety of different two-component BEC dynamics which can occur after a pulse has been stopped. Because of the initial spatial structure of created with the USL pulses, we see some novel effects in the ensuing dynamics.

Upon the switch-on, in many cases the probe will be revived via (12). We label the spatial profile of the revived probe by . When we are in the weak probe limit ( or, stated in terms of the wavefunctions, ), (12) becomes

(13) |

allowing us to make a correspondence between the revived probe and if and are known. The revived pulse will then propagate out of the BEC to the PMT at . In the absence of any attenuation or distortion during the propagation out, the spatial features of the revived probe get translated into temporal features. Thus if we observe the output then we would deduce that revived pulse was

(14) |

where is the time it takes the probe pulse to travel from some point to . Note that we are relying here on the fact that we can switch the coupling field onto its full value of with a time scale fast compared to the pulse delay times . A slow ramp up would lead to a more complicated relationship between and .

Combining (13) and (14) shows how the phase and amplitude information that was contained in at the time of the switch-on is transferred to the output probe . This transfer will be imperfect for three reasons. First, when sufficiently small spatial features are in , then is comparable to , giving rise to a significant . The resulting absorptions will cause deviations of from our expectation (13). Second, we mentioned how spatial features are translated into temporal features on the output probe. During the output, fast time features on the output probe will be attenuated via the bandwidth effect discussed in (5), affecting the accuracy of the correspondence (14). Finally, stronger output probes, which will occur when at the time of the switch-on, make both the writing at the switch-on and the subsequent propagation out more complicated and less reliable. Our following examples will demonstrate these considerations.

### iv.2 Formation and writing of interference fringes

Interesting dynamics occur in when the two internal states are trapped equally () and the scattering lengths are slightly different. We consider a case with Rb-87 atoms and choose , , and . The two lower states and are magnetically trapped with nearly identical magnetic moments, and we consider a trap with and assume a tranvserse area . These two states have an anomalously small inelastic collisional loss rate inelastic and so have been successfully used to study interacting two-component condensates for hundreds of milliseconds multipleComponentJILA . The elastic scattering lengths are

Figure 4(a) shows the initial wavefunctions after the input and stopping of a probe pulse. We see a Gaussian shaped density profile in reflecting the input probe’s Gaussian shape. The subsequent dynamics are governed by Eqs. (II) with . In this case, a weak probe () was input. As a result, is nearly constant in time and the evolution of is governed by essentially linear dynamics, with a potential determined by the magnetic trap and interactions with atoms:

(15) |

Figure 4(b) shows this potential in this case. The hill in the middle has a height and arises because atoms feel a stronger repulsion from the condensate in when they are in . Figures 4(c-d) show the subsequent dynamics. One sees that the condensate is pushed down both sides of the potential hill and spreads. However, once it reaches the border of the BEC, it sees sharp walls from the trap potential. Even for the fairly moderate scattering length difference here, there is sufficient momentum acquired in the descent down the hill to cause a reflection and formation of interference fringes near the walls. The wavelength of the fringes is determined by this momentum.

What happens if one switches the coupling field back on after these dynamics? Figures 4(e) shows the revived probe pulses upon switch-ons at the times corresponding to Fig. 4(d). One sees a remarkable transfer of the sharp density and phase features of the condensate onto the probe field, according to (13). Because we are in the weak probe regime, the coupling field intensity only very slightly deviates from it’s input value . Figures 4(f) then shows the output probe . The sharp interference fringes are able to propagate out, though there is some attenuation and washing out of the features. Note that the features at more positive propagate out first, leading to a mirror image-like relationship between the spatial Fig. 4(e) and temporal Fig. 4(f) patterns as predicted by (14).

In this particular example, we successfully output many small features from the to the output probe field. Here the fringes are , which is still larger than the absorption length , but not substantially so, leading to some small amount of dissipation during the switch-on and output. This gives us a sense of the “information capacity”. The number of absorption lengths, or optical density, which in this case is , ultimately limits the number of features (for a given desired fidelity) which could be successfully written and output. Note also that in the case, some of the amplitude occupies the entering region leading to additional imperfections in the writing process in this region.

### iv.3 Breathing behavior and long storage

The dynamics in the previous example are quite dramatic, but are not particularly conducive to preserving or controllably processing the information in the BEC. For this, it would be preferable to switch the roles of and . Such a case is shown in Fig. 5. Because in this case, the potential hill is turned into a trough (see Fig. 5(a)). The effective potential in the region of the condensate, in the Thomas-Fermi limit, becomes and so it harmonic, with a much smaller oscillator frequency than the magnetic trap. The evolution can be easily calculated by decomposing the wavefunction into a basis of the harmonic oscillator states of this potential. In the example here, there is significant occupation of the first several oscillator levels, and so one sees an overall relative phase shift in time (from the ground state energy of the zeroeth state), a slight dipole oscillation (from occupation of the first excited state) and breathing (from the second). After one oscillator period (310 ms in the case shown), replicates its original value at the switch-off.

In this case, the dynamics are quite gentle, and so the spatial scales of the phase and density features are always quite large compared with the absorption length . The fidelity of the writing and output of the information on the probe field (shown in Figs. 5(d-e)) is correspondingly better than in the previous example. Additionally, because the is trapped near the BEC center, this case avoids problems associated with occupying the region .

Because of both the ease of analyzing the evolution and the high fidelity of outputting the information, we expect this case to be well-suited to controlled processing of optical information. For example, if the input pulse created a wavefunction corresponding to the ground state of the oscillator potential, the evolution would result only in a homogenous phase shift, proportional to the storage time, allowing long storage of the information or introduction of controllable phase shifts. By choosing the pulse lengths differently so several oscillator states are occupied, one could achieve linear processing or pulse reshaping.

Furthermore, note that in this example there is a small but discernable dip in the density at 51 ms, indicating some nonlinearity in the evolution of . One can tune this nonlinearity by varying the probe to coupling ratio, leading to nonlinear processing of the information.

### iv.4 Strong probe case: Two-component solitons

The two-component dynamics are even richer when one uses strong probe pulses so nonlinear effects become very evident in the evolution. In such a case, the qualitative features of the dynamics will be strongly effected by whether or not the relative scattering lengths are in a phase separating regime phaseSeperation ; phaseSeperationTh . Experiments have confirmed that the studied in our previous examples in Rb-87 are very slightly in the phase separating regime multipleComponentJILA . Relative scattering lengths can also be tuned via Feshbach resonances feshbach .

Figure 6 shows the evolution of the two components following the input and stopping of a stronger probe (). In this case we chose our levels to be , and , which would require an optical trap opticalTrap in order to trap the and equally. This level scheme is advantageous in the strong probe case since which, as we will discuss in more detail in Section V.3, improves the fidelity of the writing and output processes. We chose the scattering length , higher than the actual background scattering length, to exaggerate the phase separation dynamics. One sees in Fig. 6(a),(c) that over a 30 ms timescale, the phase separation causes the density in to become highly localized and dense. This occurs because the scattering length causes atoms to be repelled from the region occupied by the atoms, and in turn the atoms find it favorable to occupy the resulting “well” in the density. These two processes enhance each other until they are balanced by the cost of the kinetic energy associated with the increasingly large spatial derivatives and we see the formation of two-component (vector) solitons twoCompSolitons . In the case here, two solitons form and propagate around the BEC, even interacting with each other. The alternating grey and white regions along each strip in Fig. 6(a) indicate that the solitons are undergoing breathing motion on top of motion of their centers of mass. Fig. 6(b) shows that the total density profile varies very little in time. It is the relative densities of the two components that accounts for nearly all the dynamics.

We found solitons formed even in only slightly phase separating regimes (). The number of solitons formed, the speed of their formation, and their width were highly dependent on this ratio as well as the number of atoms in . The ability of stopped light pulses to create very localized two-component structures seems to be a very effective method for inducing the formation of vector solitons, which has hitherto been unobserved in atomic BECs. A full exploration of these dynamics is beyond our scope here.

These non-trivial features can also be written onto probe pulses as shown in Fig. 7(a). Note that the condition for coherent revivals does not depend on the weak probe limit. Therefore, the writing process is still primarily a coherent process, and (12) is still well satisfied. However, we see in Fig. 7(a), the coupling field is strongly affected by the writing and is far from homogenous, meaning the weak probe result (13) can not be used. Nonetheless, we see that the qualitative features of both the density and phase of have still been transferred onto . The writing in the strong probe regime will be studied in more detail in Section V.

In Fig. 7(b) we plot the resulting output probe pulse in this case. As in the weak probe case there is some degradation in the subsequent propagation due to attenuation of high frequency components. In addition, unlike the weak probe case, there are nonlinearities in the pulse propagation itself which causes some additional distortion. Section V.3 also addresses this issue. Even so, we again see a very clear signature of the two solitons in the both the intensity and phase of the output probe.

We have performed some preliminary simulations in traps with weaker transverse confinement and which calculate the evolution in the transverse degrees of freedom. They show that the solitons can break up into two-component vortex patterns via the snake instability snake ; SolitonVortex .

We chose our parameters to be in the phase separating regime here. Just as in the weak probe case, other regimes will lead instead to much gentler dynamics. For example, it has been shown breatheTogether that when then there exist “breathe-together” solutions of the BEC whereby complete overlap of the wavefunctions persists. In this case the nonlinear atomic interaction can lead to spin-squeezing spinSqueezing . It would be extremely interesting to investigate a probe-revival experiment in such a case, to see to what extent the squeezed statistics are written into probe pulse, producing squeezed light squeezedLight . This would require taking both the light propagation and atomic dynamics in our formalism beyond the mean field. Steps in this direction have been taken in quantumProcessing ; squeezedLightRaman , but these analyses are restricted to the weak probe case.

## V Quantitative study of writing and output

The above examples demonstrate that a rich variety of two component BEC dynamics can occur, depending on the relative interaction strengths and densities of the states involved. They also show that remarkably complicated spatial features in both the density and phase can be written into temporal features of a probe light pulse and output. The purpose of this section is to quantify the fidelity with which wavefunctions can be written onto the probe field.

To do this we consider an example of a two-component BEC with a Gaussian shaped feature in , with parameters characterizing length scales and amplitudes of density and phase variations. Note these Gaussian pulse shapes are relevant to cases in which one might perform controlled processing, e.g. Fig. 5. Switching on a coupling field then generates and outputs a probe pulse with these density and phase features. We will calculate this output, varying the parameters over a wide range. In each case, we will compare the output to what one expect from an “ideal” output (without dissipation or distortion), and calculate an error which characterizes how much they differ. Using the analysis of the switching process in Section III and the USL propagation in Section II we also obtain analytic estimates of this error.

In the weak probe case, we find a simple relationship between and the output . We find that to optimize the fidelity in this case one should choose the Gaussian pulse length between two important length scales. The shorter length scale is determined by EIT bandwidth considerations, which primarily contribute error during the output. On the other hand, for large pulses comparable to the total condensate size, the error during the writing process dominates due to partially occupying the condensate entering edge region ().

In the strong probe case, we find that the relationship between and is more complicated, however, when , one can still make an accurate correspondence. We find the fidelity in this case depends only weakly on the probe strength. Conversely, when , even a small nonlinearity causes additional phase shifts and distortions, making this correspondence much more difficult, leading to higher errors for stronger probes. For this reason, a system that uses, for example (as in Fig. 7) would be preferable when one is interested in applications where there are comparable densities in the two states.

By considering the switch-on and output processes, starting with an arbitrary initial two-component BEC, we emphasize that this method of outputting the atomic field information onto light fields works regardless of how the BEC state was generated. One could prepare a BEC in a coherent superposition of and by any available method including, but not limited to, inputting a slow light pulse.

### v.1 Deriving from the written and output probe field.

Assume we have a two component BEC with wavefunctions (and again define