Superradiant phases of a quantum gas in a bad cavity

Superradiant phases of a quantum gas in a bad cavity

Simon B. Jäger Theoretische Physik, Universität des Saarlandes, D-66123 Saarbrücken, Germany    John Cooper JILA, National Institute of Standards and Technology and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440, USA Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA    Murray J. Holland JILA, National Institute of Standards and Technology and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440, USA Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA    Giovanna Morigi Theoretische Physik, Universität des Saarlandes, D-66123 Saarbrücken, Germany
August 4, 2019
Abstract

We theoretically analyse superradiant emission of light from an ultracold gas of bosonic atoms confined in a bad cavity. The atoms dipolar transition which couples to the cavity is metastable and is incoherently pumped, the atomic motion is affected by the mechanical forces of the cavity standing-wave. By means of a mean-field model we determine the conditions on the cavity parameters and pump rate that lead to steady-state superradiant emission. We show that this occurs when the superradiant decay rate exceeds a threshold determined by the recoil energy, which scales the quantum atom-photon mechanical interactions. Above this threshold, superradiant emission is accompanied by the formation of matter-wave gratings that diffract the emitted photons. The stability of these gratings is warranted when the pump rate is larger than a second threshold, below which the emitted light is chaotic. These dynamics are generated by collective quantum interference in a driven-dissipative system. The evolution presents signatures of a peculiar second-order phase transition, where coherent phases of both light and matter emerge and are controlled by entirely incoherent processes.

pacs:
37.30.+i, 42.65.Sf, 05.65.+b, 05.70.Ln

Superradiance describes the collective emission of light by an ensemble of dipoles. It is a quantum interference phenomenon in the emission amplitudes and is accompanied by a macroscopic coherence within the ensemble Dicke:1954 (); Gross:1982 (). In its original formulation, Dicke considered a medium of dipoles confined within their resonance wavelength and showed that their spontaneous decay can be enhanced by the factor Dicke:1954 (). Superradiant enhancement can also be observed when a collection of atoms interact sequentially with the resonant mode of an optical cavity Kim:2018 ().

Quantum interference is typically lost due to fluctuations in the amplitude and in the phase of the dipole-field coupling. These fluctuations can be suppressed by cooling the atomic medium to ultralow temperatures Inouye:1999 (); Baumann:2010 () and/or by subwavelength localization of the scatterers in an ordered array Vogel:1985 (); DeVoe:1996 (); Clemens:2003 (); Fernandez:2007 (); Habibian:2011 (); Reimann:2015 (); Begeley:2016 (); Neuzner:2016 (). When, in contrast, the coherence length of the atomic wave function extends over several wavelengths, e.g., when the atoms form a Bose-Einstein condensate, superradiant scattering of laser light can manifest through the formation of matter-wave gratings Inouye:1999 (); Schneble:2003 (); Piovella:2001 (); Nagy:2010 (); Baumann:2010 (). In free-space, superradiant gain can be understood as the diffraction of photons from the density grating of the recoiling atoms, which acts as a amplifying medium Inouye:1999 (); Schneble:2003 (). Within an optical resonator, these dynamics can give collective-atomic-recoil lasing Bonifacio:1994 (); Bonifacio:1994:2 (); Slama:2007 (); Bux:2013 () and can be understood as synchronization of the atomic motion induced by superradiant photon scattering Cube:2004 (); Slama:2007 ().

Figure 1: (a) A Bose-Einstein condensate is confined within a standing-wave resonator, which emits photons at rate . (b) The metastable atomic transition couples to the cavity mode and is incoherently pumped at rate . (c) The atomic gas forms density gratings after the first superradiant decay. (d) For sufficiently large values of one grating is mechanically stable and the emitted field is coherent (the field is in the reference frame of the cavity frequency).

In this Letter we analyse the interplay between superradiant emission and quantum fluctuations due to the recoiling atoms, when the atoms’ dipolar transitions couple to the mode of a lossy standing-wave resonator. In contrast to Refs. Inouye:1999 (); Schneble:2003 (); Piovella:2001 (); Nagy:2010 (); Baumann:2010 (), here the atoms are incoherently pumped, as shown in Fig. 1, and therefore no coherence is established by the process pumping energy into the system. The system parameters are chosen to be in the regime where stationary superradiant emission (SSR) is expected Meiser:2009 (); Meiser:2010:1 (); Meiser:2010:2 (); Bohnet:2012 (); Tieri:preprint (). This corresponds to the buildup of a macroscopic dipole that acts as a stationary source of coherent light. We include the mechanical effects of that light on the atomic motion, thus fluctuations occur in the atomic positions due to photon emission. We predict that SSR can result from a subtle interplay between noise and quantum fluctuations.

Consider a gas of ultracold atomic bosons with mass  that are confined along the axis of a standing-wave resonator. The atoms do not interact directly; their relevant electronic degrees of freedom form a metastable dipole with excited state  and ground state . The dipoles are incoherently pumped at rate  and strongly coupled to a cavity mode with wave number  and loss rate . The evolution of the density matrix  for the cavity field and the atoms’ internal and external degrees of freedom is governed by the Born-Markov master equation . Here, is the total kinetic energy, with the momentum of each atom . Furthermore describes the reversible evolution that arises from the interaction with the resonator, with and the annihilation and creation operators of a cavity photon, and  the cavity detuning from the atomic transition frequency. The field couples with strength  to the collective dipole , where is the th dipole lowering transition and the sum is weighted by the value of the cavity standing wave mode at the positions . The incoherent dynamics is described by the Lindbladians For a large number of atoms the quantum dynamics is numerically intractable due to the adverse Liouville space scaling. Below we argue that this model can be reduced by a mean-field model under assumptions valid in experimentally existing parameter ranges.

We first assume that  is the largest rate among the set of system parameters. In this regime (i) the atomic transition is radiatively broadened by the coupling with the cavity and its linewidth at an antinode is . (ii) The cavity field follows adiabatically the atomic motion and the cavity field operator is approximately given by the macroscopic dipole, Xu:2016 (); Jaeger:2017 (), while shot noise fluctuations are negligible Habibian:2013 (). This allows one to reduce the dynamics to the atoms’ Hilbert space with density matrix . The resulting system’s evolution is governed by a master equation , with and where describes the strong-long-range interactions mediated by cavity photons. Now the incoherent processes are the incoherent pump at rate  and the superradiant decay from the ensemble with rate . We have neglected retardation effects of the cavity field, which is justified by the choice of large . We have also neglected single-atom radiative decay at rate , assuming time scales and . Since , this time scale can be stretched to in a thermodynamic limit where is kept constant Jaeger:2017 (). Under these assumptions, when initially the many-body state is a product state, the system is well described by a mean-field treatment. For this purpose we consider the single-particle density matrix obtained by tracing out  atoms. Applying the same procedure to the master equation for , we obtain:

(1)

In this open quantum system, the incoherent evolution is due entirely to the incoherent pump. The interactions with the resonator are described by the mean-field Hamiltonian,

(2)

where we have defined . Here, the Rabi frequency is proportional to the mean-field order parameter , which is itself proportional to the intracavity field amplitude.

Figure 2: Contour plot of the rate of the first superradiant emission as a function of the incoherent pump rate (in units of ) and of the atomic gas temperature (in units of ). The solid line separates the regime in which the atoms undergo superradiant decay from the one where thermal fluctuations suppress quantum interference (stripes).

We first use this model to analyse the effect of thermal fluctuations on superradiant emission. We prepare the system in the initial (thermal) state , with the inverse temperature and the partition function. Here, . We determine the dynamics for short time by means of a stability analysis as a function of and , the details of which are reported in the Supplemental Material (SM) SM (). We identify when is stable to small fluctuations, and when instead it exponentially increases as with rate . This rate, , gives the decay rate of the first superradiant emission. Figure 2 shows the contour plot of the exponent  as a function of both and . It displays a threshold temperature above which thermal fluctuations suppress quantum interference. Below this threshold superradiant emission occurs for a finite interval of the pump rate . The maximal value, , increases with , and for it converges to . Even though this result concerns the initial dynamics of the system prepared in , it is still of interest to compare these inequalities with the interval of values of  for which SSR is expected. The buildup of a collective dipole requires that exceeds the single-atom decay rate Meiser:2010:1 (); Tieri:preprint (): In our case, the minimal value follows from our assumption that rates of the order are neglected. For SSR in a medium localized below a wavelength, it is clear that should not exceed , namely, the linewidth of the collective dipole Meiser:2010:1 (); Tieri:preprint (). Interestingly, the upper bound in Fig. 2, coincides with the one predicted by a semiclassical calculation for this setup and a homogeneous medium Jaeger:2017 ().

For the rest of this Letter, we focus on the zero-temperature limit, where the atoms form a Bose-Einstein condensate (BEC). In order to investigate the role of quantum fluctuations in determining the onset of SSR we observe that the BEC momentum state is scattered by photon absorption and emission into a family of momentum states , where their energy, , is an integer multiple of the recoil frequency .

We analyse the asymptotic behavior of Eq. (1) (which is here strictly defined in the thermodynamic limit) by means of a recursive procedure, see SM SM (). We identify three phases as a function of and , which we depict in Fig. 3(a) and denote by (i) incoherent, (ii) coherent, and (iii) chaotic. In the incoherent phase only the solution with is stable and collective effects are suppressed. In the coherent phase there is one stable solution with . As visible in the phase diagram, the condition for the appearance of this phase is that the superradiant linewidth should exceed a minimum value determined by the recoil frequency, with . Finally, in the chaotic phase both solutions with and are unstable. This phase is found when , but simultaneously the pump rate is below a certain value .

We have verified these predictions by numerically integrating Eq. (1) on the momentum family . Our initial condition was that each internal state was prepared in the excited state with momentum . Figure 3(b) displays the evolution of the absolute value of the order parameter  for different values of and for , which are the parameters of Path A in subplot (a). The intracavity field first grows exponentially and thus reaches the maximum at a time scale of the order , after which its behavior depends on the value of . For it oscillates about a finite asymptotic value. This is the coherent phase, which we also identify with SSR. We have verified that the dynamics are accompanied by the formation of matter-wave gratings at momenta , which constructively interfere with the condensate. For , instead, the oscillations are damped down to zero, corresponding to the incoherent regime. This effect is accompanied by the formation of a statistical mixture of the states and , which leads to the emission of incoherent light. We estimate the dividing line separating the incoherent from the coherent phase by determining where the coherent transition amplitude coupling the states  and  is equal to their energy offset , i.e., , so as to allow those states to be populated. Using Eq. (1) and from the semiclassical theory Jaeger:2017 (), we find leading to the estimate , which qualitatively agrees with our numerical result.

Figure 3(c) shows the details of the evolution along Path B of the phase diagram, namely at a fixed value of , by varying across the “chaotic” region. The main figure displays the contour plot of the Fourier transform of the order parameter as a function of and the inset its corresponding evolution. The transition from regular oscillations to chaos is observed at a critical value , where subharmonics appear. The oscillation frequency for is given by (in the reference frame of the cavity field) and thus increases monotonically with the incoherent pump rate. Its value can be extracted by imposing the stationary condition and assuming that kinetic and potential energy are constant in the asymptotic long-time limit SM (). Taking leads to the relation , which coincides with the numerical result. This is the regime where a stable density grating is formed. For the density grating becomes unstable and the system jumps back and forth between a prevailing occupation of the set of states corresponding to an even grating, , and of the set of states corresponding to an odd grating, . While the states within each set are coupled by coherent processes, the two sets are only coupled to each other by the incoherent pump. When the coherent coupling rate exceeds the pump rate , the system performs quantum jumps from one spatial order (and a corresponding coherent field) to another. In Fig. 3(c) the value of is shown for which the stability analysis predicts the transition from chaotic to incoherent dynamics. Here, the spectrum is dense and it is not possible to identify a clear threshold.

Figure 3: (a) Phase diagram in the plane indicating the phases of the system, characterized by the properties of the emitted light, and identified by solving Eq. (1) at the asymptotic dynamics SM (). Path A and B show the parameters of subplot (b) and (c), respectively. (b) Contour plot of the dynamics of the mean-field order parameter as a function of time (in units of ) and (in units of ), for and . The data is obtained by integrating numerically Eq. (1) with , and the atoms initially prepared in the excited state at momentum . (c) Contour plot of the Fourier transform (in arbitrary units) as a function of (in units of ) and of (in units of ), for and , obtained by integrating Eq. (1) over the interval until (the inset shows the corresponding dynamics of ). The vertical dashed line shows the value ; the top horizontal line gives the pumping rate , where a bifurcation occurs, the circles indicate the frequencies found by means of a stability analysis SM (). The bottom horizontal line shows the value that corresponds to the expected border between chaotic and coherence phase of subplot (a).
Figure 4: (a) Overlap and (b) stationary value of the mean-field order parameter (see text) as functions of (in units of ). The symbols are the data points of the numerical integration of Eq. (1) (‘o’ and ‘+’ correspond to integration times , and and to momentum intervals and , respectively). The dashed curve in (a) shows Eq. (3), and in (b), it is the stationary value of obtained by setting . The inset in (b) shows the stationary value of (in units of ) as a function of (in units of ). The parameters are and .

We now focus on the transition from the incoherent to the coherent phase by varying the parameters along Path A of Fig. 3(a). For this purpose we consider the state , which is the statistical mixture of two gratings and is formed after the first superradiant emission followed by a repump cycle. The state has no finite dipole moment and therefore the order parameter vanishes. The weight corresponds to the probability that the first superradiant emission has occurred so that the atomic wave function can form a grating at wavelength (the corresponding quantum state is ). We determine its value with the stability analysis of Eq. (1) giving:

(3)

Imposing we recover and therefore deduce the second constraint , which is qualitatively consistent with the threshold value separating the incoherent from the coherent phase. Figure 4(a) shows the overlap between the state and the stationary state of Eq. (1) as a function of for (along Path A): The discontinuity in the derivative of the overlap at indicates the transition to SSR. The dotted line indicates the probability of Eq. (3), which agrees with the numerical results for . Subplot (b) displays the corresponding stationary value  of the order parameter. We determine this value from the asymptotic solution of Eq. (1) (see Ref. SM ()) and also by numerical integration of the dynamics of , taking . The two results qualitatively agree and show that exhibits a transition at from zero to a finite value. An analysis of the eigenvalues of the partial transpose of the density matrix shows that internal and external degrees of freedom become entangled at the transition point. The quantum fidelity, moreover, exhibits a discontinuity at SM (). These features are indicative of a second order phase transition. The formation of a grating should be accompanied by localization of the atom at the minima, and thus to an increase of the kinetic energy, as visible in the inset of Fig. 4(b). The coherent phase that emerges is solely controlled by and , and thus by a complex interplay between quantum interference, noise, and dissipation.

These dynamics require that the superradiant linewidth exceeds a threshold value determined by the recoil frequency. They can be observed when the dipole decay rate is smaller than the recoil frequency, for instance in quantum gases of alkali-earth metals whose intercombination line couples to the resonator Norcia:2016 (); Gothe:2017 (). The corresponding signatures can be monitored by detecting the coherence properties of the emitted light.

Acknowledgements.
The authors are grateful to L. Giannelli, F. Rosati, S. Schütz, and M. Xu for discussions. This work has been supported by the German Research Foundation (DACH “Quantum crystals of matter and light”), the German Ministry of Education and Research (BMBF) within the Quantera project NAQUAS, the European Commission (ITN ColOpt). and by the German Ministry of Education and Research (BMBF) via the Quantera project “NAQUAS”. Project NAQUAS has received funding from the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union’s Horizon 2020 Programme.

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Supplemental Material: Superradiant phases of a quantum gas in a bad cavity

s.0.1 Stability analysis

In this section, we study the stability of a stationary state of the mean-field master equation

(1)

The explicit form of is given in Eq. (1) of the main article. The stability of a stationary state , with , is determined by the initial dynamics of a density matrix with a small perturbation Campa:2009 (). If this perturbation is amplified over time we can state that is unstable, otherwise is stable.

Using the mean-field master equation (1) we derive an equation of motion for that takes the form

(2)

Here we have defined and used the definition

(3)

with and . In Eq. (2), we have included only first order perturbations in and we have discarded the second order. Applying the Laplace transform we derive the following equation

with

(4)

The entries of the matrix take the form

(5)
(6)
(7)
(8)

Inverting and applying the inverse Laplace transformation, we obtain the dynamics of . To calculate the dynamics we need to know the poles when we invert the matrix . These are roots of the dispersion relation

(9)

The complex solution of Eq. (9) with the largest real part gives the dominant contribution to the dynamics of . Therefore this determines whether the stationary solution is stable or not. If the perturbation will exponentially grow and thus is an unstable stationary solution. Otherwise, if , is stable.

s.0.2 Asymptotic state

In this section we explain how we calculate the stationary state of the system leading to the diagram in Fig. 3(a) of the main article.

A significant class of stationary states are given by incoherent states

(10)

These are referred to as incoherent since the collective dipole vanishes. These states are stationary if they commute with the kinetic energy . Although these states do not show a collective dipole they can be used to calculate the onset of superradiance. This is accomplished in the next section. In this section we explain how we find stationary states that show a non-vanishing collective dipole .

We will show that there is a stationary state where and is not time dependent while oscillates with a constant frequency in time.

Using Eq. (1) one can show that

(11)

Explicitly denoting the amplitude and phase we obtain

(12)

Now assuming that there exists a stationary state with and we arrive at

(13)

Therefore to find a stationary solution for the system we need to solve

(14)

This is equivalent to calculating the stationary state in the frame oscillating with the frequency shown in Eq. (13).

To characterize and numerically determine this solution we use the order parameter . For the numerical calculation of the stationary state we start from an order parameter and find to recalculate the new value of . We iterate this step until and converges.

In the case when there is a solution of Eq. (14) with we know that there is a coherent stationary state. However, this state does not need to be stable. To calculate the stability we use the dispersion relation in Eq. (9).

s.0.3 Stability of the incoherent state

The aim of this section is to describe the stability of the incoherent state in Eq. (10). This method leads to the stability diagram of a thermal state visible in Fig. 2 of the main article. Moreover we derive Eq. (3) of the main article that leads to the predicted line in Fig. 4(a).

Since we observe that the matrix in Eq. (4) becomes diagonal. Therefore if we want to find the zeros of the dispersion relation in Eq. (9) it is sufficient to solve the equation

(15)

Using the duality of the Schrödinger and Heisenberg pictures we obtain

(16)

Here we use the definition that for an operator the expectation value is defined as .

In the homogeneous case, we calculate , and it takes the form

(17)

with . Using Eq. (17) in Eq. (16) we obtain

(18)

where we explicitly used the fact that all particles are in the excited state and therefore holds. From the identity

and momentum translation

we can show that

(19)

Here it is necessary that the condition can only hold for . This is true since needs to commute with that is a stationary state. Using

(20)

and Eq. (19) we get

(21)

with .

Figure 2 of the main article shows for a thermal state as function of and . The value of is found by solving numerically Eq. (15) using Eq. (21).

In what follows we calculate for a state of the form

(22)

where , and for and . The obtained result will be used to derive Eq. (3) of the main article. The quantity defined in Eq. (21) takes the form

(23)

The last term in Eq. (23) is due to coherences in and that are present in the state in Eq. (22). We see that in the limit where is small with respect to the momentum width this term is usually negligible. However, if the quantized structure of the momentum states become important this term will contribute to the stability of the state. In the limit where we can expand Eq. (23) and get

(24)

Using Eq. (15) the value of takes the form

(25)

From the fact that we obtain

(26)

The stability line can then be derived by solving . We obtain the critical pumping strength

(27)

If we expect that the distribution is unstable. Equation (27) shows that in zeroth order in only contributes to the stability of the state. The next order is governed by the contribution of the coherences in and . It is negative and therefore lowers the critical pumping strength . This means that the coherences stabilize the incoherent state.

For the special case discussed in the main article we use and to derive Eq. (3) in the main article.

s.0.4 Fidelity and Entanglement

To analyse the behavior of the system close to the transition we calculated the fidelities and . They are defined as

(28)
(29)

with and Horodecki:2009 (). Here we define as the steady state that is reached by the simulations for each corresponding value of .

The two fidelities and are different. The fidelity is equal to one if and only if , whereas for the fidelity this is only true if both density matrices are pure states. Therefore also measures the purity of the states.

The fidelities in Eq. (28) and Eq. (29) are shown in Fig. SLABEL:Fig:5 (a) and (b), respectively. They are calculated for different values of and . The black dots correspond to the simulations that were performed with a cut-off at and the gray crosses for the simulations with . Both are in very good agreement.

As is visible in Fig. SLABEL:Fig:5(a) is almost equal to for values . For the fidelity decreases rapidly to a value of around and then for increases again to . This behavior shows the rapid change of the properties of the density matrix at the transition point . In Fig. SLABEL:Fig:5(b) the dip that is observed in Fig. SLABEL:Fig:5(a) in the interval to is also visible but much less pronounced.

Remarkably, as is increased and before shows the local minimum in the interval , it first peaks at what can be interpreted as a point at which the purity is maximal. For smaller and larger we observe only lower values of .

Another property of the system is the entanglement between the internal and external degrees of freedom which is a peculiar feature associated with the formation of a matter-wave grating supporting SSR. As already explained in the main article we observe that for the system remains incoherent. In that case at steady state all particles are in the excited state and therefore the system has no entanglement between internal and external degrees of freedom. However, if the system is in the coherent phase, for , we expect that the system shows entanglement between internal and external degrees. This claim can be verified by an analysis of the partial transpose of the stationary state Horodecki:2009 (). The matrix is calculated from by applying the transpose on the internal degrees of freedom only. In the case where internal and external degrees of freedom are not entangled is a positive matrix. On the other hand if is not positive we know that internal and external degrees must be entangled. To check whether the internal and external degrees of freedom are entangled in the coherent phase we calculate the minimum eigenvalue

(30)

If this eigenvalue is negative we know that the system is entangled. The numerical calculated values for are shown in Fig. S2.

Figure S2: The value of as defined in Eq. (30) as a function of (in units of ). The parameters and markers are chosen as in Fig. SLABEL:Fig:5.

As expected we observe that there is no entanglement for . For we observe a negative , demonstrating that in this region, the stationary state is an entangled state.

References

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