Propagation of a quantum fluid of light in a cavityless nonlinear optical medium: General theory and response to quantum quenches
Making use of a generalized quantum theory of paraxial light propagation where the radiation-axis and the temporal coordinates play exchanged roles, we discuss the potential of bulk nonlinear optical media in cavityless configurations for quantum statistical mechanics studies of the conservative many-body dynamics of a gas of interacting photons. To illustrate the general features of this point of view, we investigate the response of the fluid of light to the quantum quenches in the photon-photon interaction constant experienced at the front and the back faces of a finite slab of weakly nonlinear material. Extending the standard Bogoliubov theory of dilute Bose-Einstein condensates, peculiar features are predicted for the statistical properties of the light emerging from the nonlinear medium.
pacs:42.65.-k, 03.70.+k, 42.50.Lc, 47.37.+q
After several decades during which the study of systems of many interacting particles has focused on matter fluids such as liquid helium, electron gases in solid-state materials, ultracold atom vapors, nuclear-matter fluids, or quark-gluon plasmas in colliders, photon propagation in suitably designed nonlinear optical systems is presently attracting a growing interest as a novel platform to investigate the physics of interacting Bose gases, the so-called quantum fluids of light (see Ref. Carusotto2013 for a review). The interactions between the photons constituting the fluid of light are mediated by the Kerr () optical nonlinearity of the underlying medium.
So far, numerous experimental observations have been performed for semiconductor-planar-microcavity geometries, including the demonstration of a Bose-Einstein-like condensation Kasprzak2006 , of a superfluid flow Amo2009 , and of the hydrodynamic nucleation of nonlinear excitations such as solitons Amo2011 and quantized vortices Nardin2011 ; Sanvitto2011 in dilute photon gases. In the meanwhile, active investigations have addressed the possibility of realizing systems characterized by very large optical nonlinearities. Correspondingly, the induced photon-photon interactions are expected to introduce strong quantum correlations within the fluid of light, generating in turn new quantum phases.
In this quest Carusotto2013 ; Houck2012 ; Chang2014 , researchers have been faced to (at least) two main difficulties. On the one hand, obtaining strong enough nonlinearities in scalable systems to study the dynamics of a strongly interacting photon gas in a spatially extended system turns out to be a major experimental challenge. On the other hand, the dynamics of a light field in devices based on cavities is intrinsically a driven-dissipative one, which introduces severe complications in the theoretical description of such systems and which is typically very detrimental for the study of purely quantum features.
An alternative platform for studying many-body physics in photon fluids is based on light propagating in a bulk nonlinear (of Kerr type) optical medium. From elementary classical optics, it is well known that the paraxial propagation of a spectrally narrow beam of light can be described within the so-called paraxial and slowly-varying-envelope approximations (see, e.g., Refs. Agrawal1995 ; Raghavan2000 ; Rosanov2002 ; Boyd2008 ) by a nonlinear wave equation formally analogous to the Gross-Pitaevskii equation for the order parameter of a dilute Bose-Einstein condensate Pitaevskii2003 . As originally pioneered in Refs. Lai1989a ; Lai1989b , this framework naturally translates upon quantization to a many-body quantum nonlinear Schrödinger formalism with the roles of the propagation coordinate and time exchanged.
This framework has been used in a number of theoretical works where laser-physics problems have been reformulated in the hydrodynamics language Mattar1981 , including, e.g., the investigation of superfluid-like behaviors in the flow of a photon fluid Pomeau1993 ; Hakim1997 ; Leboeuf2010 ; Carusotto2014 ; Larre2015 , of nonlinear phenomena with light waves Dekel2007 ; Khamis2008 ; Dekel2009 ; Cohen2013 , and of the so-called acoustic Hawking radiation Fouxon2010 ; Fleurov2012 ; BarAd2013 ; Vinish2014 . From the experimental point of view, numerous works have been devoted to the study of nonlinear features that may appear in these systems, with a special attention dedicated to their relation to hydrodynamics and superfluidity aspects Vaupel1996 ; Wan2007 ; Jia2007 ; Wan2010a ; Wan2010b ; Jia2012 ; Vocke2015 . A major first step in the very quantum direction of realizing a gas of strongly interacting photons in a propagating geometry has been recently reported using an optically dressed atom gas in the Rydberg-EIT regime as a bulk nonlinear medium: This has allowed for the experimental observation of strongly repulsing photons Peyronel2012 and, soon after, of two-photon molecular bound states Firstenberg2013 . These remarkable experimental advances call for a theoretical approach that is able to describe in its full generality the many-body dynamics of strongly interacting photons propagating in a cavityless configuration.
Building atop the pioneering theoretical works Lai1989a ; Lai1989b and Refs. Wright1991 ; Crosignani1995 ; Hagelstein1996 ; Kolobov1999 ; Matsko2000 ; Tsang2006 , we report here a fully general quantum field theory of the propagating photon fluid. In this approach, the roles played by the optical-axis coordinate and the time parameter are exchanged: Light propagation in the direction is naturally described in terms of evolution equations while the direction corresponds to a third spatial dimension in addition to the transverse and directions. In a paraxial-propagation configuration, light diffraction provides an effective mass to the photons in the plane and chromatic dispersion leads to a—typically different—effective mass in the direction. As usual, the Kerr nonlinearity of the medium gives rise to photon-photon interactions. In contrast to microcavity configurations where driving and dissipation play a major role in the photon-fluid dynamics Carusotto2013 , in paraxial-propagation geometries, the quantum fluid of light follows a fully conservative Hamilton dynamics starting from an initial condition determined by the incident light field and its coherence properties. The quantum state of the optical field at the end of the evolution is experimentally accessible via a measurement of the statistical properties of the transmitted light emerging from the nonlinear optical medium.
As a most remarkable example of application of this formalism, we then study the transmission of a coherent light across a finite slab of weakly nonlinear medium. In this very simple configuration, the photons experience a pair of sudden jumps of the interaction parameter upon crossing the front and the back faces of the nonlinear medium. As a result of these two quantum quenches, the fluid of light gets excited and its quantum state after the conservative evolution across the nonlinear material can be reconstructed from the statistical properties of the transmitted light. In the weak-nonlinearity regime, the main excitation process consists in the emission of pairs of correlated counterpropagating Bogoliubov phonons, which reflects in peculiar features in the intensity distribution and in the near- and the far-field two-body correlation functions. In its simplicity, this example illustrates the power of the conservative propagation dynamics in view of generating, detecting, and manipulating strongly correlated quantum phases of matter in photon gases, as well as of studying quantum dynamical features of many-body systems Kinoshita2006 ; Polkovnikov2011 .
The paper is organized as follows. First of all, in Sec. II, we review the classical propagation equation of a paraxial beam of light in a cavityless nonlinear optical medium of Kerr type. On this basis, we present in Sec. III a general theory that makes it possible to describe the evolution of the quantum optical field for generic values of the nonlinear interaction parameter. In Sec. IV, we discuss how the relatively small fluctuations superimposing upon a coherent-light field in the weak-nonlinearity regime can be treated within the framework of the Bogoliubov theory of dilute Bose-Einstein condensates. As an application of this formalism, we investigate in Sec. V the two-body quantum correlations resulting from the propagation of a laser beam across a slab of weakly nonlinear medium and interpret their features in terms of a dynamical Casimir emission of Bogoliubov collective excitations in a temporally modulated quantum fluid of light. Finally, in Sec. VI, we draw our conclusions and give prospects to the present work.
Ii Classical wave equation
We consider the propagation of a laser wave in a dispersive and inhomogeneous Kerr dielectric for which the (frequency-dependent) electric susceptibility reads [one writes down , with ]
where the homogeneous contribution takes into account the chromatic dispersion of the medium, the linear modulation comes from the existence of spatial inhomogeneities or of an optical confinement, and is the Kerr nonlinear shift of the susceptibility, proportional to the local electric intensity , i.e., to the square modulus of the (complex representation of the) electric field. For simplicity’s sake, the dielectric is assumed to be devoid of free charge carriers and nonmagnetic. We finally suppose that the optical field maintains its polarization in the course of its propagation in the medium so that a scalar approach is valid. As explained, e.g., in Ref. Rosanov2002 , this is possible for paraxial beam of light and provided slowly varies on space scales of the order of the optical wavelength. Spin-orbit-coupling effects resulting from significant deviations from the optical axis and significant spatial variations of the electric susceptibility will be subjected to a future publication LarreFuture .
Introducing the envelope of the laser-wave electric field oscillating at the angular frequency as
where “c.c.” stands for “complex conjugate,” and making use of the standard paraxial and slowly-varying-envelope approximations (see, e.g., Refs. Agrawal1995 ; Raghavan2000 ; Rosanov2002 ; Boyd2008 ), the Maxwell equations supplemented by Eq. (1) lead to the following classical wave equation for :
In this equation, denotes the nabla operator in the plane and the functions and are defined as , where ; the parameters , , and are respectively the propagation constant ( is the vacuum speed of light) of the laser wave in the direction, the group velocity of the photons in the medium, and the group-velocity dispersion evaluated at the carrier’s angular frequency .
The hydrodynamic interpretation of the propagation equation (3) is mostly well known in the limiting case of a purely monochromatic wave at Pomeau1993 ; Hakim1997 ; Dekel2007 ; Khamis2008 ; Dekel2009 ; Leboeuf2010 ; Cohen2013 ; Carusotto2014 ; Larre2015 and has offered a transparent physical interpretation to a number of nonlinear-optics experiments Vaupel1996 ; Wan2007 ; Jia2007 ; Wan2010a ; Wan2010b ; Jia2012 ; Elazar2012 ; Elazar2013 ; Vocke2015 . In this time-independent case, the first and second derivatives of the envelope with respect to vanish, in such a way that the propagation of the optical field decribed by Eq. (3) recovers the mean-field dynamics of the order parameter of a dilute two-dimensional Bose-Einstein condensate, playing the role of some external potential and corresponding to the effective two-dimensional boson-boson interaction constant.
On the other hand, only a very few studies so far Lai1989a ; Lai1989b have investigated the consequences of this analogy in time-dependent regimes, where, by extension, the propagation of the photon fluid in the positive- direction can be written in terms of a time evolution in a three-dimensional space where the physical time parameter plays the role of a third spatial coordinate in addition to the transverse variables and . In the following, a special attention will be paid to the new hydrodynamic features that originate from this dependence.
In order to facilitate () to be viewed as a true time (space) parameter, we introduce the following coordinates:
respectively homogeneous to a time and a length. In these new variables, the paraxial-propagation equation (3) takes the form of a time-dependent Gross-Pitaevskii equation:
where the electric-field envelope now has to be considered as a function of and and as functions of . This dependence of and corresponds in our language to a temporal dependence, which, as we shall see in the following, opens the way to the study of quantum-quench physics in the framework of paraxial optics. On the other hand, the fact that the medium properties do not depend on the physical time implies that and are independent on the spatial coordinate.
In the Gross-Pitaevskii-like equation (5), the rigid-drift (in the direction) term originates from the group velocity of the photons in the medium and the kinetic operator
involves a contribution in the (actual-time) direction in addition to the usual one in the plane. The anisotropy of the “mass” tensor
in Eq. (6) is a natural consequence of the different origins of the effective masses , in the (spatial) , directions and in the (temporal) one: The former are due to diffraction while the latter originates from dispersion; note that in vacuum, and so is infinite, by definition.
When the medium is characterized by an anomalous group-velocity dispersion, that is, when , one has and so the mass matrix is positive in all the directions. In the following (see Sec. IV for a detailed discussion), we shall see that the dynamical stability of the fluid of light requires such a negative , but also repulsive photon-photon interactions, which sets and therefore , that is, that the Kerr nonlinearity is self-defocusing.
Iii Quantum theory
In order to describe the quantum features of a light beam propagating in the nonlinear medium considered in Sec. II, the classical photon field verifying the paraxial wave equation (5) must be replaced with a quantum field operator satisfying suitable boson commutation relations. To this purpose, in the present work, we perform a canonical quantization Dirac1930 ; Weinberg1995 of the classical field theory from which Eq. (5) may be derived. The procedure first consists in rewriting the evolution equation (5) in Lagrangian and then Hamiltonian form (Sec. III.1). Section III.2 is dedicated to the precise determination of the global multiplicative constant appearing in the Lagrangian and the Hamiltonian of the paraxial-propagation problem; while it plays no role at the classical level, it starts having a crucial importance upon quantization. The quantization is finally accomplished in Sec. III.3 by replacing the conjugate fields of the classical Hamiltonian theory with quantum field operators obeying equal- bosonic commutation relations, standardly deduced from the canonical Poisson-bracket relations.
Of course, similar quantized wave equations describing paraxial light propagation in nonlinear optical media have been considered in the past (see, e.g., Refs. Lai1989a ; Lai1989b ; Wright1991 ; Matsko2000 ) and applied to concrete problems related to quantum soliton propagation Crosignani1995 ; Hagelstein1996 : As a crucial addition to these works, we do not restrict our attention to one-dimensional fiber geometries for which only the coordinate matters LarreFutureBis , but fully take into account the dynamics of the optical field in the transverse plane. Furthermore, an explicit expression for the normalization constant appearing in the boson commutators is provided. In contrast to works on the quantization of paraxial electromagnetic fields Deutsch1991 ; Aiello2005 , our approach is able to naturally describe spatiotemporal light propagation and to include the possible existence of spatial inhomogeneities and/or of an optical confinement as well as of an optical nonlinearity. As we shall see later, all these features will be crucial for the theory to be applicable to the problem considered in Sec. V, which requires an accurate description of the peculiar spatial, angular, and spectral correlations displayed by the laser beam emerging from the back face of the nonlinear medium.
At this stage, the reader who is just interested in knowing the basic elements structuring the quantum theory in order to apply the latter to concrete quantum-optics problems can skip Secs. III.1 and III.2 and go directly to Sec. III.3, where one finds the commutation relations of the quantum field operator associated to the classical photon field [Eqs. (26)], the many-body quantum Hamiltonian of the paraxial-propagation problem [Eq. (27)], and the corresponding Heisenberg evolution equation [Eq. (28)].
iii.1 Hamiltonian formulation
Let us introduce the Lagrangian
with the Lagrangian density
It is immediate to notice that the Euler-Lagrange equations of motion and deduced from Eqs. (8) and (9) coincide with the evolution equation (5) and its complex conjugate, respectively. The global normalization factor in the definition (9) of the Lagrangian density will be rigorously determined in Sec. III.2 on the basis of microscopic calculations. It plays obviously no role at the classical level but is on the contrary crucial to derive the exact commutation relations of the quantum field operators.
Equation (5) being a first-order differential equation in , the data of the electric-field envelope at a given time is sufficient to determine the subsequent evolution of the system. As a result, the Lagrangian , functional of , , and their derivatives with respect to , contains an overabundant number of dynamical variables. Thus, before searching for the conjugate momenta and moving on to the Hamiltonian formalism, it is convenient to eliminate the redundant dynamical variables from the Lagrangian. Following Ref. CohenTannoudji1989 , one starts by rewriting the Lagrangian density as a function of , , and their spatiotemporal derivatives. In particular, the term involving and in Eq. (9) reads . Thus, by adding to , we get a Lagrangian which does not depend on and which is by definition equivalent to . Its density reads
By means of the Euler-Lagrange equation relating to , that is, , one may express as a function of and its time derivative. Inserting this expression of into , we finally obtain a Lagrangian which only involves the dynamical variable and its time derivative .
Now let us denote by the conjugate momentum of . By definition,
Along the extremal “path,” , and as a consequence, . According to Eq. (10), this yields
As conjugated fields, and obey the canonical relations
where is the equal- Poisson bracket.
iii.2 Normalization constant
Before moving on to the canonical quantization of the classical field theory (9, 15), let us determine the multiplicative constant in terms of the optical parameters of the electromagnetic wave. was introduced as a global factor in Eq. (9) to ensure that has the dimension of an energy per unit volume, as it has to be according to Eq. (8). While such a normalization constant plays no role at the classical level, since it cancels out in the Euler-Lagrange equations of motion, it becomes important upon quantization as it determines the actual spacing between the energy levels of the system. The role of the quantized action in the Bohr-Sommerfeld quantization rules is the most well-known examples of such a dependence.
A possible strategy to fix is to determine the total energy carried by the laser wave of complex electric field [; see Eq. (2)] in the simple case where and , and compare the latter with the well-known formula of classical electrodynamics for the time-averaged energy of a quasimonochromatic electromagnetic field propagating through a dispersive, homogeneous, linear, nonmagnetic medium, that is (see, e.g., Ref. Landau1960 ),
In Eq. (16), denotes the (frequency-dependent) permittivity of the medium, being the one of free space, , where is the slowly varying envelope of the magnetic field (deduced, e.g., from Maxwell-Faraday’s law), and is the vacuum permeability.
In order to get an expression for the physical energy in our Lagrangian formalism, one has to perform a Legendre transformation of the Lagrangian with respect to the actual time coordinate, , instead of the “propagation one,” . As the Jacobian determinant and since the action of the optical system can be alternatively defined in the coordinates as , the Lagrangian density (9) stays invariant under the coordinate transformations (4): . Thus, one finds from Eq. (9) that the conjugate momentum  of  in the coordinates is given by
as a consequence of which the Hamiltonian density associated to in the coordinates reads—when and —as
Making the substitution into Eq. (18), one deduces the Hamiltonian as a functional of the total electric field :
The temporal evolution of the envelope is given by the Hamilton equation of motion , being the equal- Poisson bracket—not to be confused with the equal- Poisson bracket defined in Sec. III.1. Inserting into this evolution equation, one ends up with the following equation for the total field :
where, making use of ,
Within the framework of the paraxial and slowly-varying-envelope approximations, one can check that defined in Eq. (23) gives a negligible contribution to the total energy of the electromagnetic wave. By comparing Eqs. (16) and (22), one finally obtains the normalization constant in terms of the angular frequency and the propagation constant of the laser in the medium:
Brief discussions on the constant normalizing the Lagrangian and the Hamiltonian in one-dimensional fiber geometries were given in Ref. Lai1989a and, in a bit more detailed way, in Ref. Wright1991 . A physical interpretation of expression (24) will be given in the next section.
iii.3 Canonical quantization
In order to carry out the canonical quantization of the (classical) Hamiltonian theory presented in Sec. III.1, one replaces the conjugated fields and with quantum field operators and (by choice, in the Heisenberg picture) and the Poisson bracket with the commutator , where is the reduced Planck constant. On doing so, Eqs. (13) become
from which and thanks to Eq. (12) one deduces the following equal-time, that is, equal-, commutation relations:
being the Hermitian conjugate of : .
It is worth noting that, instead of reducing the overabundant number of dynamical variables in the Lagrangian in order to avoid dealing with fields which are not independent from each other in the transition from the Lagrangian formalism to the Hamiltonian one and in the canonical quantization via the Poisson bracket, we could also have implemented the more basic and robust Dirac-Bergmann quantization procedure Dirac1930 ; Sundermeyer1982 to pass from the classical theory to the quantum one, as recently used in Ref. Vinish2014 . As detailed in footnote NoteDiracTheory , Dirac’s procedure produces the same equal-time commutation relations for the quantum field operators as the ones deduced from the canonical quantization method adopted in this work, which fully validates our approach.
From the definition , one notes that the commutation rules (26) involve quantum field operators at the same “propagation” time but at different physical times and . Such a writing is based on the coordinate transformation , which is of course legitimate only in the paraxial and slowly-varying-envelope approximations and if back-scattered waves are assumed not to exist, i.e., if light propagation is assumed to occur only in the positive- direction. Such a quantization procedure imposing equal- and different- (instead of equal- and different-) canonical commutation relations was pioneered in Refs. Lai1989a ; Lai1989b and critically discussed in Ref. Matsko2000 . It is also important to insist on the fact that the slow spatiotemporal variation condition that is assumed for the classical electric-field envelope directly applies to its quantum counterpart : A discussion of such a condition in a quantum framework is illustrated in Refs. Deutsch1991 ; Aiello2005 .
By inserting the explicit expression (24) of the normalization parameter into the commutation rule (26a), one easily gets that the multiplicative constant in the right-hand side of Eq. (26a) reads , yielding a simple interpretation of in terms of the energy density of a single-photon wavepacket of total energy .
Equation (27) corresponds to the many-body quantum Hamiltonian describing the evolution in time of a many-photon laser beam propagating through the bulk inhomogeneous and nonlinear optical medium of electric susceptibility (1). In the dielectric, the position of a point is referenced by the coordinates and . The two first contributions are the kinetic terms in the transverse plane and in the direction with different effective masses (as discussed in Sec. II), the second line describes the rigid global drift along the axis due the group velocity of light in the medium, and the two last terms respectively account for the spatial modulation of the electric susceptibility and for the two-photon interactions mediated by the Kerr nonlinearity of the dielectric.
In the theory of ultracold Bose fluids, contact interactions are usually considered in place of the actual—but much more complicated—two-body interactions. This approximation is very helpful in simplifying the many-body quantum problem and is well accurate as long as the inter-particle distance is much larger than the range of the boson-boson interactions (see Ref. Pitaevskii2003 ). Here, the assumed local form of the Kerr optical nonlinearity automatically leads to contact-like interactions between the photons of the light beam, that is, no dilutness condition for the photon gas is required to get the four-field interaction term in Eq. (27). Since no hypothesis is made on the intensity of the photon-photon interaction parameter , the many-body Hamiltonian (27) can describe a quantum fluid of weakly interacting photons, that is, in the Gross-Pitaevskii/Bogoliubov regime, as well as a strongly interacting one. In what follows, we will focus our attention on the weak-nonlinearity regime, in which the quantum fluctuations of the fluid of light can be described within the framework of the well-known Bogoliubov theory of dilute Bose-Einstein condensates (see, e.g., Refs. Pitaevskii2003 ; Castin2001 ; Fetter2003 ). Motivated by the intense experimental investigations that are presently in progress Peyronel2012 ; Firstenberg2013 , the strong-interaction regime will be the subject of future works Lebreuilly ; LebreuillyBis .
For the sake of completeness, it is useful to explicitly write the evolution equation of the operator . In the Heisenberg picture, it is obtained from the system’s Hamiltonian as . Using Eq. (27) and taking advantage of the commutation relations (26), this gives
Equation (28), which governs the time evolution of in the space, is simply the quantized version of the classical equation (5). As originally pointed out in Ref. Lai1989a for a one-dimensional waveguide geometry, it has the form of a quantum nonlinear Schrödinger equation.
Iv Bogoliubov theory of quantum fluctuations
iv.1 General framework
In an illuminated dielectric medium devoid of free charges as the one considered in this paper, quantum noise of the electromagnetic field only arises from the quantum uncertainty of the optical field, that is, in more physical terms, from the discreteness of the photon. In the case of a strongly coherent light beam propagating across a weakly nonlinear three-dimensional bulk medium, quantum noise is typically small and can be described in terms of weak-amplitude quantum fluctuations oscillating on top of a strongly classical wave.
Mutuating well-known results from the theory of weakly interacting ultracold atomic gases Pitaevskii2003 ; Castin2001 ; Fetter2003 , one may develop a Bogoliubov-like theory based on an expansion of the envelope operator of the form
In this expression, the classical field , which satisfies the Gross-Pitaevskii-like equation (5), corresponds to the coherent component of the electric-field envelope and is a small quantum correction to . As the whole quantum nature of the optical field is captured in the fluctuation operator , the equal- commutation relations (26) then totally transfer to the latter, giving
Linearizing the Heisenberg equation of motion (28) with respect to and its Hermitian conjugate, one readily gets the so-called Bogoliubov-de Gennes equation
which encodes the time evolution of the quantum fluctuation in the three-dimensional space and is in fact the heart of the Bogoliubov approach. With respect to similar (classical) equations considered in the literature Carusotto2014 ; Vinish2014 ; Vocke2015 , this equation explicitly includes the dependence of the field and the corresponding effective mass, given by the group-velocity dispersion at .
In Sec. IV.2, we review the Bogoliubov theory of linearized fluctuations in position- and time-independent configurations. Although well established and known in the context of matter fluids, it is important to quickly review it within the nonlinear-propagating-geometry context because (i) of the nontrivial role of the effective mass in the dynamics of the luminous fluid in the temporal direction and (ii) of the existence of on-going experiments which aim at probing the phononic part of the Bogoliubov excitation spectrum of a propagating fluid of light Vocke2015 ; Biasi .
iv.2 Spatially homogeneous system
In the simplest case where the classical background field is at some time homogeneous in all the (, , and ) directions, with , and when the nonlinear optical medium is spatially homogeneous with a constant Kerr coefficient ,
In such a configuration, the electric-field envelope follows a simple harmonic evolution with a linearly-evolving (in time ) global phase: , where . In the theory of dilute Bose-Einstein condensates, the wavenumber correponds to the chemical potential of the Bose gas Pitaevskii2003 .
In the homogeneous situation (32), the elementary excitations of the fluid of light are plane waves of wavevector in the space, the solution of the Bogoliubov-de Gennes equation (31) obeying the mode expansion
The mode operators satisfy the same- commutation relations
and evolve harmonically as
with a frequency determined by the -dependent wavenumber
The so-called Bogoliubov amplitudes and [ and independent in the situation (32)] are finally given by
By definition, they satisfy the normalization condition . Because of the photon-photon interactions (), is nonzero: This indicates that the ground state of the Bogoliubov theory differs from the trivial vacuum without particles in the modes Pitaevskii2003 .
Modulo the contribution to the oscillation pulsation , originating from the presence of the drift term in the Bogoliubov-de Gennes equation (31), the wavenumber defined in Eq. (36) corresponds to the well-known Bogoliubov dispersion relation for the elementary excitations propagating on top of a uniform dilute Bose-Einstein condensate at rest, the quadratic function given by Eq. (37) playing the role of the single-particle kinetic energy of matter-wave superfluids. Note that this result stems from the conservative nature of the considered dynamics and is in contrast to the rich variety of dispersions predicted for driven-dissipative fluids of light in microcavity architectures Carusotto2013 . The graphical representation of is given in Fig. 1 for repulsive photon-photon interactions, i.e., for , in the two cases .
In the anomalous-dispersion case, that is, when , the photon effective mass in the direction is—as the photon effective mass in the transverse plane—positive, and the “kinetic energy” stays as a consequence positive for any . If one also considers a self-defocusing nonlinearity, that is, if , the Bogoliubov dispersion relation never acquires an imaginary part, which means that the photon-photon collision processes mediated by the underlying nonlinear medium do not give rise to unstable behaviors in the photon fluid.